User:Jianing Song

You don't try to extend these sequences: A000679, A094268, A257479.

Some really fast-growing functions in the OEIS: A056041/A266203 (the weak Goodstein function, growth $\omega$), A154714 ($\{f_n(2)\}$, growth $\omega$), A188545 (the "fuse" function, growth ${\epsilon }_0$), A028444/A060843 (the "BB" function, growth ${\omega }_1^{\mathrm{C}\mathrm{K}}$).

Note that $fuse(3)>2\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 16$, and $BB(6)>2\upuparrows 2\upuparrows 2^{2^{2^{2^{2^{2^{2^{2^6}}}}}}}$ (see here).

The most interesting sequences in my eyes: A046073, A105876, A298821.

The sequences I submitted myself that I'm most satisfied with: A306198, A400743.

Theta series for some best-known lattices

Name	Description	Theta series $\{a(n)\}$	$\{a(n)/a(1)\}$
Square	$\mathbb{\mathbb{Z}}[i]$	A004018	A002654
Hexagonal, A2	$\mathbb{\mathbb{Z}}[\omega ]$	A004016	A002324
Cubic	${\mathbb{\mathbb{Z}}}^3$	A005875	-
4D cubic	${\mathbb{\mathbb{Z}}}^4$	A000118	A046897
D4	${\mathbb{\mathbb{Z}}}^4\cup {(\mathbb{\mathbb{Z}}+\frac{1}{2})}^4$	A004011	A000593

Examples of factorizations

For $G$ a subgroup of $\mathrm{Gal}(L/K)$, we write $L^G$ to be the fixed field of $G$. We consider $L/K=\mathbb{\mathbb{Q}}({\zeta }_{24})/\mathbb{\mathbb{Q}}$, with Galois group $(\mathbb{\mathbb{Z}}/24\mathbb{\mathbb{Z}}{)}^{\times }$.

Fixed field	Field	Discriminant	${\mathbb{\mathbb{Q}}}_2\otimes ...$	$(2)\subset \mathbb{\mathbb{Z}}$	${\mathbb{\mathbb{Q}}}_3\otimes ...$	$(3)\subset \mathbb{\mathbb{Z}}$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,5\}}$	$\mathbb{\mathbb{Q}}(\mathrm{i},\sqrt{6})$	2304	${\mathbb{\mathbb{Q}}}_2(\mathrm{i},\sqrt{6})$	${\mathfrak{𝔭}}_2^4$	${\mathbb{\mathbb{Q}}}_3({\zeta }_{12})$	${\mathfrak{𝔭}}_3^2$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,7\}}$	$\mathbb{\mathbb{Q}}(\sqrt{2},\sqrt{-3})$	576	${\mathbb{\mathbb{Q}}}_2(\sqrt{2},\sqrt{-3})$	${\mathfrak{𝔭}}_2^2$	${\mathbb{\mathbb{Q}}}_3({\zeta }_{12})$	${\mathfrak{𝔭}}_3^2$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,11\}}$	$\mathbb{\mathbb{Q}}(\sqrt{-2},\sqrt{3})$	2304	${\mathbb{\mathbb{Q}}}_2(\sqrt{-2},\sqrt{3})$	${\mathfrak{𝔭}}_2^4$	${\mathbb{\mathbb{Q}}}_3(\sqrt{3}{)}^2$	${\mathfrak{𝔭}}_3^2\mathfrak{𝔭}{\text{'}}_3^2$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,13\}}$	$\mathbb{\mathbb{Q}}({\zeta }_{12})$	144	${\mathbb{\mathbb{Q}}}_2({\zeta }_{12})$	${\mathfrak{𝔭}}_2^2$	${\mathbb{\mathbb{Q}}}_3({\zeta }_{12})$	${\mathfrak{𝔭}}_3^2$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,17\}}$	$\mathbb{\mathbb{Q}}({\zeta }_8)$	256	${\mathbb{\mathbb{Q}}}_2({\zeta }_8)$	${\mathfrak{𝔭}}_2^4$	${\mathbb{\mathbb{Q}}}_3(\mathrm{i}{)}^2$	${\mathfrak{𝔭}}_3\mathfrak{𝔭}{\text{'}}_3$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,19\}}$	$\mathbb{\mathbb{Q}}(\sqrt{-2},\sqrt{-3})$	576	${\mathbb{\mathbb{Q}}}_2(\sqrt{-2},\sqrt{-3})$	${\mathfrak{𝔭}}_2^2$	${\mathbb{\mathbb{Q}}}_3(\sqrt{-3}{)}^2$	${\mathfrak{𝔭}}_3^2\mathfrak{𝔭}{\text{'}}_3^2$
$\mathbb{\mathbb{Q}}({\zeta }_{24}{)}^{\{1,23\}}$	$\mathbb{\mathbb{Q}}(\sqrt{2},\sqrt{3})$	2304	${\mathbb{\mathbb{Q}}}_2(\sqrt{2},\sqrt{3})$	${\mathfrak{𝔭}}_2^4$	${\mathbb{\mathbb{Q}}}_3({\zeta }_{12})$	${\mathfrak{𝔭}}_3^2$

Chebyshev's bias

Subgroup	Coset	Primes	Coset	Primes	Coset	Primes
$\{1,5\}$	$\{7,11\}$		$\{13,17\}$		$\{19,23\}$
$\{1,7\}$	$\{5,11\}$		$\{13,19\}$		$\{17,23\}$
$\{1,11\}$	$\{5,7\}$		$\{13,23\}$		$\{17,19\}$
$\{1,13\}$	$\{5,17\}$	-1: A297354/A297355	$\{7,19\}$	-1: A297356/A297357 0: A380877	$\{11,23\}$	very large
$\{1,17\}$	$\{11,19\}$	very large	$\{5,13\}$	-1: A297447/A297448 0: A379989 differences: A379731	$\{7,23\}$	-1: A295353/A295354
$\{1,19\}$	$\{5,23\}$	-1: A398234	$\{7,13\}$	-1: A398235	$\{11,17\}$	-1: A398236
$\{1,23\}$	$\{5,19\}$		$\{7,17\}$		$\{11,13\}$

The world of nuclides is full of mysteries

Unknown decays: ⁴⁸Ca (β⁻), ¹²³Te (β⁺), ¹⁴⁹Sm (α), ^180mTa (β⁺/β⁻/IT), ¹⁸⁷Os (α), ²²²Fr (α), ²⁴⁷Cm (β⁻), ²⁴⁸Bk (α/β⁺/β⁻);

Unknown if decay is energetically possible: β⁻ of ²²²Rn.

Bound-state ${\beta }^{-}$ decays

By conservation of energy, we have $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{n+})=Q_{{\beta }^{-}}({}_Z^A\mathrm{X})+\sum _{i=1}^n{I}_i(Z)-\sum _{i=1}^n{I}_i(Z+1)$, where $I_i(Z)$ and $I_i(Z+1)$ are respectively the i-th ionization energy of element X and the next element.

Write $E(Z)=\sum _{i=1}^Z{I}_i(Z)$ be the total binding energies of electrons, then

$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-n)+})=Q_{{\beta }^{-}}({}_Z^A\mathrm{X})+(E(Z)-\sum _{i=1}^n{I}_{Z+1-i}(Z))-(E(Z+1)-\sum _{i=1}^{n+1}{I}_{Z+2-i}(Z+1))=Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+1)-E(Z))+\sum _{i=1}^n(I_{Z+2-i}(Z+1)-I_{Z+1-i}(Z))+I_{Z+1-n}(Z+1)$.

In particular, for $n=0,1$, we have $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-n)+})\approx Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+1)-E(Z))+I_{Z+1}(Z+1)$;

for $n=2,\cdots ,9$, we have $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-n)+})\approx Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+1)-E(Z))+I_{Z-1}(Z+1)$.

First category: $Q_{{\beta }^{-}}<0$ for neutral atoms

Data from NIST (please beware that these are most predictions). All values are in unit of keV.

Beta decay	$Q_{{\beta }^{-}}({}_Z^A\mathrm{X})$	$E(Z+1)-E(Z)$	$I_{Z+1}(Z+1)$	$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{Z+})$	${\sum }_{i=1}^2(I_{Z+2-i}(Z+1)-I_{Z+1-i}(Z))+I_{Z-1}(Z+1)$	$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-2)+})$
148Eu → 148Gd	－28 ± 10	11.6	59.06554	19	18.08746	－22
163Dy → 163Ho	－2.831 ± 0.022	12.5	65.13713	49.806	19.84548	4.514
193Ir → 193Pt	－56.63 ± 0.30	15.9	90.65984	18.13	27.2647	－45.27
194Au → 194Hg	－28 ± 4	16.8	95.89819	51	28.7966	－16
202Tl → 202Pb	－40 ± 4	17.3	101.3367	44	30.3914	－27
205Tl → 205Pb	－50.6 ± 0.5	17.3	101.3367	33.4	30.3914	－37.5
213Po → 213At	－74 ± 5	18.4	109.8872	17	32.9085	－59
215At → 215Rn	－88 ± 9	18.7	112.8422	6	33.7745	－73
222Rn → 222Fr	－6 ± 8	19.2	115.8575	91	34.6741	9
244Pu → 244Am	－73.1 ± 2.7	22	142.1535	47.1	42.5273	－52.6
243Am → 243Cm	－6.9 ± 1.6	24	145.7401	114.8	43.621	12.7
246Bk → 246Cf	－120 ± 60	23	153.124	10	45.845	－100

Second category: $Q_{{\beta }^{-}}>0$ for neutral atoms

Here we only consider the nuclides with $Q_{{\beta }^{-}}({}_Z^A\mathrm{X})<200\mathrm{k}\mathrm{e}\mathrm{V}$. There are two cases:

$\bullet$ There are excited states of the daughter with energies lying between $Q_{{\beta }^{-}}({}_Z^A\mathrm{X})$ and $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{Z+})$, thus providing more channels to decay when the parent is fully ionized: ¹⁸⁷Re, ¹⁹⁴Os, ²²⁷Ac, ²⁴¹Pu, ²⁴⁷Cm, ²⁵⁰Cm, ²⁴⁹Bk.

$\bullet$ There are no such excited states, so full ionizing only increases beta-decay energies of the original paths: ²¹⁰Pb, ²¹²At, ²²²Rn^*, ²²⁸Ra.

Beta decay	$Q_{{\beta }^{-}}({}_Z^A\mathrm{X})$	$E(Z+1)-E(Z)$	$I_{Z+1}(Z+1)$	$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{Z+})$	${\sum }_{i=1}^2(I_{Z+2-i}(Z+1)-I_{Z+1-i}(Z))+I_{Z-1}(Z+1)$	$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-2)+})$
187Re → 187Os* (9.756 keV)	－7.289 ± 0.002	15.2	85.61442	63.125	25.7921	3.303

See here, Table IV for details.

^* It is not clear which category should ²²²Rn be put into.

Bound-state $2{\beta }^{-}$ decays

Similarly, we have $Q_{2{\beta }^{-}}({}_Z^A{\mathrm{X}}^{n+})=Q_{2{\beta }^{-}}({}_Z^A\mathrm{X})+\sum _{i=1}^n{I}_i(Z)-\sum _{i=1}^n{I}_i(Z+2)$, hence

$Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-n)+})=Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+2)-E(Z))+\sum _{i=1}^n(I_{Z+3-i}(Z+2)-I_{Z+1-i}(Z))+I_{Z+2-n}(Z+2)+I_{Z+1-n}(Z+2)$.

In particular, for $n=0$, we have $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{Z+})\approx Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+2)-E(Z))+2I_{Z+2}(Z+2)$;

for $n=1$, we have $Q_{{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-1)+})\approx Q_{{\beta }^{-}}({}_Z^A\mathrm{X})-(E(Z+2)-E(Z))+I_{Z+2}(Z+2)+I_Z(Z+2)$.

Double beta decay	$Q_{2{\beta }^{-}}({}_Z^A\mathrm{X})$	$E(Z+2)-E(Z)$	$I_{Z+2}(Z+2)+I_{Z+1}(Z+2)$	$Q_{2{\beta }^{-}}({}_Z^A{\mathrm{X}}^{Z+})$	$(I_{Z+2}(Z+2)-I_Z(Z))+I_{Z+1}(Z+2)+I_Z(Z+2)$	$Q_{2{\beta }^{-}}({}_Z^A{\mathrm{X}}^{(Z-1)+})$
152Sm → 152Gd	－55.7	22.96	116.84945	38.2	75.85972	－2.8
164Dy → 164Er	－25.0	25.2	133.08972	82.9	86.29105	36.1
214Po → 214Rn	－150	37.1	223.6886	37	144.6083	－42
242Pu → 242Cm	－86.8	46	289.051	156.3	186.9038	54.1

Nuclides with the lowest mass among isobars of mass numbers 141 ~ 209

For 141 ≤ A ≤ 209, the nuclide with the lowest mass among isobars of A (¹⁴¹Pr, ¹⁴²Nd, ¹⁴³Nd, ¹⁴⁴Nd, ¹⁴⁵Nd, ¹⁴⁶Sm, ..., ²⁰⁹Bi) are of particular interest. The following table lists their predicted half-lives here:

A	Nuclide	log Tα (yr)
146	146Sm	7.96
147	147Sm	11.03
186	186Os	15.30
144	144Nd	15.36
148	148Sm	15.80
187	187Os	17.66
149	149Sm	18.47
151	151Eu	18.66
209	209Bi	19.30
176	176Hf	20.51
177	177Hf	22.00
145	145Nd	22.93
192	192Pt	22.96
178	178Hf	23.64
185	185Re	25.02
188	188Os	26.28
150	150Sm	28.05
191	191Ir	29.09
170	170Yb	29.24
189	189Os	31.18
179	179Hf	31.31
182	182W	32.82
175	175Lu	35.28
184	184W	35.97
204	204Pb	36.01
171	171Yb	36.50
181	181Ta	38.78
183	183W	39.46
172	172Yb	42.34
194	194Pt	44.47
180	180Hf	45.84
190	190Os	47.39
169	169Tm	47.63
198	198Hg	52.15
154	154Gd	52.60
173	173Yb	62.37
195	195Pt	64.13
166	166Er	64.52
206	206Pb	66.55
193	193Ir	67.00
197	197Au	71.78
174	174Yb	75.31
203	203Tl	80.57
167	167Er	80.96
196	196Pt	82.67
199	199Hg	83.82
143	143Nd	88.12
168	168Er	91.90
200	200Hg	95.46
153	153Eu	>100
160	160Dy	>100
152	152Sm	>100
155	155Gd	>100
156	156Gd	>100
208	208Pb	>100
161	161Dy	>100
157	157Gd	>100
207	207Pb	>100
159	159Tb	>100
158	158Gd	>100
165	165Ho	>100
162	162Dy	>100
201	201Hg	>100
163	163Dy	>100
142	142Nd	>100
202	202Hg	>100
164	164Er	>100
205	205Tl	>100
141	141Pr	>100

For nuclides in the table with predicted alpha half-lives longer than 10¹⁰⁰ years (including being stable to alpha decay), cluster decay modes (for example ¹⁶O emission) must be taken into account when modeling the half-lives. It seems that there is no concensus on predicting cluster decay half-lives, in contrast to alpha decay.

Warning: The following table is purely insane and contains nothing serious.

My guess of cluster decay half-lives

Nuclide	log Tα (yr)	log T12C (yr)	log T14C (yr)	log T18O (yr)	log T24Ne (yr)	minimum
141Pr	-	245.75	-	210.85	291.50	210.85
205Tl	283.07	201.02	201.97	219.02	226.19	201.02
164Dy	-	204.89	210.06	206.46	187.41	187.41
202Hg	305.64	188.90	186.02	204.49	215.28	186.02
142Nd	-	188.35	498.81	185.32	253.31	185.32
163Dy	-	193.71	202.49	190.41	181.18	181.18
201Hg	171.10	174.61	173.33	190.21	210.54	171.10
162Dy	336.47	171.38	175.29	171.00	173.99	171.00
165Ho	242.65	169.21	194.22	188.44	187.58	169.21
158Gd	-	175.63	159.36	166.90	178.65	159.36
159Tb	-	159.93	157.32	157.48	179.61	157.32
207Pb	157.17	193.02	198.06	215.52	230.38	157.17
157Gd	-	150.39	144.38	152.04	181.80	144.38
161Dy	132.24	156.10	160.63	157.78	175.85	132.24
208Pb	128.87	203.20	202.37	223.74	236.49	128.87
156Gd	-	127.89	130.31	139.36	182.19	127.89
155Gd	304.37	111.31	117.55	127.23	184.32	111.31
152Sm	164.24	115.84	111.23	127.65	187.96	111.23
160Dy	110.29	135.61	144.68	145.50	177.93	110.29
153Eu	144.69	108.55	110.99	122.63	186.92	108.55

I wish ⁴He had energy 6 MeV higher

Then α decays would essentially disappear. Warning: The following table is purely insane and contains nothing serious.

My guess of decay modes without α decay

Nuclide	Q8Be (MeV)	log T8Be (yr)	Q12C (MeV)	log T12C (yr)	Q14C (MeV)	log T14C (yr)	minimum
215At	14.84	12.03	21.89	26.09	22.13	25.62	12.03
212Rn	11.51	19.88	20.84	28.16	19.28	31.22	19.88
214Rn	14.52	12.78	23.03	23.85	22.16	25.56	12.78
216Rn	17.06	6.79	24.94	20.10	24.58	20.81	6.79
217Rn	16.33	8.51	25.95	18.11	25.91	18.19	8.51
218Rn	15.00	11.65	26.16	17.70	26.89	16.27	11.65
220Rn	13.22	15.85	23.71	22.52	28.54	13.02	13.02
222Rn	11.61	19.64	21.49	26.88	26.45	17.13	17.13
219Fr	15.53	10.40	29.65	10.84	29.42	11.29	10.40
218Ra	17.66	5.38	30.43	9.31	28.74	12.63	5.38
220Ra	15.70	10.00	32.02	6.18	31.04	8.11	6.18
221Ra	14.68	12.40	30.58	9.01	32.40	5.44	5.44
222Ra	13.85	14.36	29.05	12.02	33.05	4.16	4.16
223Ra	12.83	16.77	27.72	14.63	31.83	6.56	6.56
224Ra	12.10	18.49	26.37	17.29	30.53	9.11	9.11
226Ra	10.37	22.57	23.85	22.24	28.20	13.69	13.69
225Ac	12.30	18.02	26.87	16.30	30.48	9.21	9.21
224Th	14.80	12.12	30.36	9.45	32.93	4.39	4.39
226Th	13.04	16.27	27.67	14.73	30.55	9.07	9.07
228Th	11.22	20.56	24.99	20.00	28.20	13.65	13.65

The primary decay mode of ²²⁶Th without α decay would perhaps be ¹⁸O emission (unknown in the real world).

The primary decay mode of ²²⁸Th without α decay would be ²⁰O emission (known in the real world; the partial half-life is 1.69×10¹³ years).

The primary decay mode of ²³⁰U without α decay would be ²²Ne emission (known in the real world; the partial half-life is 1.15×10¹² years).

The primary decay mode of ²³²U without α decay would be ²⁴Ne emission (known in the real world; the partial half-life is 7.74×10¹² years).

Hands in Poker

Imagine that an online system of Poker allows players to customize the number of suits d and the number of ranks n. What should be the ranking of hands in the game?

By the way, a straight means five cards with consecutive ranks. However, the highest-ranking card can be used as high or low in a straight, and thus there are $n-3$ of them in terms of card values.

$\bullet$ For the standard deck, the card ranking is 2 3 4 5 6 7 8 9 10 J Q K A, hence the lowest-ranking straight is A2345;

$\bullet$ In Six-plus hold 'em, the card ranking is 6 7 8 9 10 J Q K A, hence the lowest-ranking straight is A6789;

$\bullet$ In the deck of Five Crowns, the card ranking is 3 4 5 6 7 8 9 10 J Q K, hence the lowest-ranking straight is K3456.

When we pick randomly 5 cards from a deck with general $n$ and $d$, the number of combinations of each hand is given as follows. A hand having a smaller number of combinations in the table ranks higher, except that a pair always beats high card (even if high card occurs less often if

$\{\begin{array}{ll}n\le 11,& d=4;\\ n\le 12,& d=5,6,7,8,9;\\ n\le 13,& d\ge 10.\end{array}$

The concrete mode of game (Texas, Omaha, or anything original) is irrelevant to the determination of ranking of hands.

Hand	Card value type	Suit combinations of hand	Card value combinations of hand
Straight flush	1+1+1+1+1	$d$	$n-3$
Five of a kind	5	$(\genfrac{}{}{0ex}{}{d}{5})$	$n$
Four of a kind	4+1	$(\genfrac{}{}{0ex}{}{d}{4})d$	$n(n-1)$
Full house	3+2	$(\genfrac{}{}{0ex}{}{d}{3})(\genfrac{}{}{0ex}{}{d}{2})$	$n(n-1)$
Flush	1+1+1+1+1	$d$	$(\genfrac{}{}{0ex}{}{n}{5})-(n-3)$
Straight	1+1+1+1+1	$d^5-d$	$n-3$
Three of a kind	3+1+1	$(\genfrac{}{}{0ex}{}{d}{3})d^2$	$\frac{n(n-1)(n-2)}{2}$
Two pairs	2+2+1	${(\genfrac{}{}{0ex}{}{d}{2})}^{2}d$	$\frac{n(n-1)(n-2)}{2}$
One pair	2+1+1+1	$(\genfrac{}{}{0ex}{}{d}{2})d^3$	$\frac{n(n-1)(n-2)(n-3)}{6}$
High card	1+1+1+1+1	$d^5-d$	$(\genfrac{}{}{0ex}{}{n}{5})-(n-3)$
Total (= $\sum$ suit combinations $\times$ card value combinations)			$(\genfrac{}{}{0ex}{}{dn}{5})$

Please note that we cannot add a hand that "contains one card in each suit" for $d\ge 5$ because, as $d\to \mathrm{\infty }$, 100% of draws of 5 cards consist of 5 cards in different suits. This is a special case of a principle when setting the list of hands: a hand is only valuable when it contains "similarities", not "differences".

Other observations should be mentioned. We always have

$\bullet$ Straight flush > Four of a kind > Full house > Straight > Three of a kind > Two pairs > One pair > High card for $d\le 4$;

$\bullet$ Five of a kind > Straight flush > Four of a kind > Full house > Straight > Three of a kind > Two pairs > One pair > High card for $d=5$; and

$\bullet$ Straight flush > Five of a kind > Four of a kind > Full house > Straight > Three of a kind > Two pairs > One pair > High card for $d\ge 5$,

but the positions of a flush and a straight depend on the specific value of $n$. A flush can rank anywhere between a straight flush and high card, while a straight can only rank below a five-of-a-kind and above a three-of-a-kind. The turning point of the positions of

$\bullet$ a flush and a full house is $n=10^{\frac{1}{3}}{d}^{\frac{4}{3}}-\frac{3}{4}\times 10^{\frac{1}{3}}{d}^{\frac{1}{3}}+3+o(1)$ as $d\to \mathrm{\infty }$;

$\bullet$ a flush and a straight is $n=120^{\frac{1}{4}}d+\frac{7}{4}+o(1)$ as $d\to \mathrm{\infty }$;

$\bullet$ evidently enough, a five-of-a-kind ranks lower than a flush if $n\le d$ and higher if $n>d$ (hence the turning point is $n=d\sim d+1$).

Turning points for specific $d$:

$\bullet d=4$: a flush and a full house change positions at $n=12\sim 13$, and a flush and a straight change positions at $n=15\sim 16$;

$\bullet d=5$: a flush and a four-of-a-kind change positions at $n=11\sim 12$, and a flush and a full house change positions at $n=16\sim 17$;

$\bullet d=6$: a flush and a four-of-a-kind change positions at $n=13\sim 14$.

Notice. Somebody may think that the ranking of hands should be determined by 7-card combinations rather than 5-card combinations, since Poker is a game of seeing 7 cards. The problem is, a 7-card combination contains in general 5-card combinations in different categories, so how to assess the value of each 7-card combination? In other words, the number of 7-card combinations of each hand can only be calculated after the determination of the ranking, not before. Since combinations of higher rankings need to be deducted while determining the number of each 7-card combination, those numbers no longer reflect the probability of getting each hand.

Standard deck ($n=13$, $d=4$)

Hand	Combinations
Straight flush	40
Four of a kind	624
Full house	3744
Flush	5108
Straight	10200
Three of a kind	54912
Two pairs	123552
One pair	1098240
High card	1302540
Total	2598960

Six-plus hold 'em ($n=9$, $d=4$)

In the game Six-plus hold 'em, there are only 9 card values used: 6 7 8 9 10 J Q K A from the lowest to the highest.

Hand	Combinations
Straight flush	24
Four of a kind	288
Flush	480
Full house	1728
Straight	6120
Three of a kind	16128
Two pairs	36288
One pair	193536
High card	122400
Total	376992

Deck of French Tarot ($n=14$, $d=4$)

The only commercially available deck of playing cards with more than 13 cards in each suit. It is difficult for people from other countries to imagine just how much the French enjoy playing tarot.

Hand	Combinations
Straight flush	44
Four of a kind	728
Full house	4368
Flush	7964
Straight	11220
Three of a kind	69888
Two pairs	157248
One pair	1537536
High card	2030820
Total	3819816

Deck of Uno No Mercy ($n=16$, $d=4$)

In a deck of Uno No Mercy each color contains 16 kinds of different cards: 0-9, Reverse, Skip, Draw 2, Draw 4, Skip Everyone, Discard All. I list this pair of $(n,d)$ here because $n=16$ is the least value such that a straight beats a flush.

Hand	Combinations
Straight flush	52
Four of a kind	960
Full house	5760
Straight	13260
Flush	17420
Three of a kind	107520
Two pairs	241920
One pair	2795520
High card	4442100
Total	7624512

Deck of Five Crowns ($n=11$, $d=5$)

The only commercially available deck of playing cards with more than 4 suits. A quite niche board game.

Hand	Combinations
Five of a kind	11
Straight flush	40
Flush	2270
Four of a kind	2750
Full house	11000
Straight	24960
Three of a kind	123750
Two pairs	247500
One pair	1650000
High card	1416480
Total	3478761

Non-uniqueness of the par contract in contract bridge

Have you noticed that in rare cases, when some suit is fixed to be the trump suit, the number of tricks for N/S (the number of tricks which can be taken by NS if either N or S is the declarer) and the number of tricks for E/W can add up to more than 13? If so, the par contract may depend on who bids first! (See here for examples!)

First let's think about the question of finding the par contract in general. Here, instead of comparing contracts that either side can make, we compare the double dummy result. I will write, say, "we have the condition C=9", to mean if club is the trump suit, then the double dummy result is that when either I and my partner is the declarer, the optimal result for us is exactly 9 tricks.

Let A be the highest-ranking condition (in the sequence C=7, D=7, H=7, S=7, NT=7, ..., C=13, D=13, H=13, S=13, NT=13) that we have, and B be the highest-ranking condition that the opponents have. Suppose WLOG that A ranks higher than or equal to B. Now we list all the conditions we have that rank at least as high as B, then left over the most valuable ones. The least to the most valuable conditions are as follows:

C=7, D=7

H=7, S=7

NT=7, C=8, D=8

H=8, S=8, C=9, D=9

NT=8

C=10, D=10

H=9, S=9

NT=9, C=11, D=11

H=10, S=10

NT=10

H=11, S=11

NT=11

C=12, D=12

H=12, S=12

NT=12

C=13, D=13

H=13, S=13

NT=13

The resulting conditions reflect the ideal contract in the bidding process if sacrifice bids are not used. I call these conditions the optimal conditions of a deal.

Now we check the double dummy result of our opponents in the suits of the optimal conditions. Pick the highest-ranking (in the sequence C=0, D=0, H=0, S=0, NT=0, ..., C=13, D=13, H=13, S=13, NT=13) condition (call it B'). If B'+1 ranks higher than B (if B' is, say, C=7, then B'+1 is the condition C=8), then the suit of B' is the key suit. If the number of tricks for N/S and the number of tricks for E/W add up to more than 13 given that the key suit is the trump suit, then it is very likely that the par contract depends on which side bids first!

See these very long tables made by me for more details.

Table of minimal polynomial of -2 cos(2π/n) (note the minus sign!)

Let P_n be the minimal polynomial of $-2\cos \frac{2\pi }{n}$ for n ≥ 3, and P₁ = P₂ := 1. Then we have:

$\prod _{d|n}{P}_d={\left(\begin{array}{cc}X& -1\\ 1& 0\end{array}\right)}_{2,1}^{n/2}$ for even n; $\prod _{d|n}{P}_d={\left(\begin{array}{cc}X& -1\\ 1& 0\end{array}\right)}_{1,1}^{(n-1)/2}+{\left(\begin{array}{cc}X& -1\\ 1& 0\end{array}\right)}_{1,2}^{(n-1)/2}$ for odd n.

PARI code:

Prod(n) = my(M = [X,-1;1,0]); if(n%2==0, (M^(n/2))[2,1], (M^((n-1)/2) * [1;1])[1,1])
P(n) = my(P = 1); fordiv(n, d, P *= Prod(n/d)^moebius(d)); P

n	Pn	Pn(k) for k = 3	k = 4	k = 6
3	$X-1$	2	3	5
4	$X$	3	4	6
5	$X^2-X-1$	5	11	29
6	$X+1$	4	5	7
7	$X^3-X^2-2X+1$	13	41	169
8	$X^2-2$	7	14	34
9	$X^3-3X-1$	17	51	197
10	$X^2+X-1$	11	19	41
11	$X^5-X^4-4X^3+3X^2+3X-1$	89	571	5741
12	$X^2-3$	6	13	33
13	$X^6-X^5-5X^4+4X^3+6X^2-3X-1$	233	2131	33461
14	$X^3+X^2-2X-1$	29	71	239
15	$X^4+X^3-4X^2-4X+1$	61	241	1345
16	$X^4-4X^2+2$	47	194	1154
17	$X^8-X^7-7X^6+6X^5+15X^4-10X^3-10X^2+4X+1$	1597	29681	1136689
18	$X^3-3X+1$	19	53	199
19	$X^9-X^8-8X^7+7X^6+21X^5-15X^4-20X^3+10X^2+5X-1$	4181	110771	6625109
20	$X^4-5X^2+5$	41	181	1121

Let p be an odd prime not dividing k² - 4, then there exists a unique number of the form $\alpha (k,p)=\frac{p-\left(\frac{k^2-4}{p}\right)}{m}\ge 3$ such that p divides P_α(k,p)(k). The parity of m is determined by the value of $\left(\frac{-(k-2)}{p}\right)$. Consider the case k = 4, so k² - 4 = 12 and -(k - 2) = -2:

p	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59	61	67	71	73	79	83	89	97
corresponding m	1	1	2	1	2	2	1	1	1	1	6	2	1	3	2	1	4	5	2	1	2	2	6
$\left(\frac{-(k-2)}{p}\right)$	-1	-1	1	-1	1	1	-1	-1	-1	-1	1	1	-1	-1	1	-1	1	-1	1	-1	1	1	1

(Also, you may find this document written by myself interesting. At the end of page 4, there is a conjecture on the densities of the occurrences of α(k,p) odd, α(k,p) ≡ 2 (mod 4), and 4 | α(k,p)).

Wall-Sun-Sun primes in Lucas sequences

Let T(P,Q,n) be the n-th term of the Lucas sequence (P,Q) of the first kind, then we have $T(k,1,n)=\frac{T(\sqrt{k-2},-1,2n)}{\sqrt{k-2}}$ for k ≥ 3. For odd primes p that are not factors of k² - 4 (so that $p-\left(\frac{k^2-4}{p}\right)$ is even), we always have $p\mid \frac{T(\sqrt{k-2},-1,p-\left(\frac{k^2-4}{p}\right))}{\sqrt{k-2}}=T(k,1,\frac{p-\left(\frac{k^2-4}{p}\right)}{2})$, so it would be meaningful to ask when p² divides this quantity. The n-Wall-Sun-Sun primes correspond to k = n² + 2.

Note the product formula $\frac{T(\sqrt{k-2},-1,n)}{\sqrt{k-2}}=\prod _{d|n,d\ge 3}{P}_d(k)$ for even n and $T(\sqrt{k-2},-1,n)=\prod _{d|n,d\ge 3}{P}_d(k)$ for odd n, where P_n is the minimal polynomial of $-2\cos \frac{2\pi }{n}$.

The OEIS has A238490 for k = 4, A238736 for k = 6 and A337791 for k = 38.

Numbering of flags in Endless mode of Plants vs. Zombies

Let $x$ (a 32-bit signed integer) be the number of the current level. Set $y=20\star x$ (32-bit signed multiplication), then:

$\bullet$ The level $x$ starts at flag $y/10$ (32-bit signed integer division) and ends at flag $y/10+2$. This will be written as flags $[y/10,y/10+2)$ in the table at the end.

$\bullet$ For $w=1,\cdots ,20$, the $w$-th flag of level $x$ has zombies if and only if $y+w=-1,0,\cdots ,1073741824$ (32-bit signed addition) or $y+w=2147483647,-2147483648,\cdots ,-1073741824$. Only Flag Zombies appear for waves not in these ranges.

$\bullet$ For Pool and Fog settings, the game crashes on a level when there are no zombies other than Flag Zombies in the 20 waves.

Let's look at $x=53687091$ for an example. We have $y=1073741820$, so we start at flag $107374182$ and end at flag $107374184$. Notice: The next level crashes with Pool or Fog because there would be no zombies. That's why many believe that the game only has 107374184 flags, which is of course not true (just try Day, Night, or Roof).

Wave $w$	1	2	3	4	5	...	20
$y+w$	1073741821	1073741822	1073741823	1073741824	1073741825	...	1073741840
Has zombies?	Yes	Yes	Yes	Yes	No	...	No

Look at also $x=107374182$. Then $y=2147483640$, so we start at flag $214748364$ and end at flag $214748366$. Warning: The next level starts at flag $-214748363$ as we plug in $x=107374183$.

Wave $w$	1	2	3	4	5	6	7	8	...	20
$y+w$	2147483641	2147483642	2147483643	2147483644	2147483645	2147483646	2147483647	-2147483648	...	-2147483636
Has zombies?	No	No	No	No	No	No	Yes	Yes	...	Yes

A summarizing table:

These waves are normal (-1~1073741824)

These waves have only Flag Zombies (game crash with Pool or Fog) (1073741825~2147483646, -1073741823~-2)

These waves are somehow normal (2147483647~-1073741824)

Period	Quarter-period	Levels	# of levels	Flags contained	Wave numbers
1	1a	0~53687091	53687092	[0,107374184)	Levels 0~53687090 (flags [0,107374182)): 1~1073741820 Level 53687091 (flags [107374182,107374184)): 1073741821~1073741824, 1073741825~1073741840
	1b	53687092~107374182	53687091	[107374184,214748366)	Levels 53687092~107374181 (flags [107374184,214748364)): 1073741841~2147483640 Level 107374182 (flags [214748364,214748366)): 2147483641~2147483646, 2147483647~-2147463636
	1c	107374183~161061273	53687091	[-214748363,-107374181)	Levels 107374183~161061272 (flags [-214748363,-107374183)): -2147483635~-1073741836 Level 161061273 (flags [-107374183,-107374181)): -1073741835~-1073741824, -1073741823~-1073741816
	1d	161061274~214748364	53687091	[-107374181,1)	Levels 161061274~214748363 (flags [-107374181,-1)): -1073741815~-16 Level 214748364 (flags [-1,1)): -15~-2, -1~4
2	2a	214748365~268435455	53687091	[0,107374182)	Levels 214748365~268435455 (flags [0,107374182)): 5~1073741824
	2b	268435456~322122547	53687092	[107374182,214748366)	Levels 268435456~322122546 (flags [107374182,214748364)): 1073741825~2147483644 Level 322122547 (flags [214748364,214748366)): 2147483645~2147483646, 2147483647~-2147483632
	2c	322122548~375809638	53687091	[-214748363,-107374181)	Levels 322122548~375809637 (flags [-214748363,-107374183)): -2147483631~-1073741832 Level 375809638 (flags [-107374183,-107374181)): -1073741831~-1073741824, -1073741823~-1073741812
	2d	375809639~429496729	53687091	[-107374181,1)	Levels 375809639~429496728 (flags [-107374181,-1)): -1073741811~-12 Level 429496729 (flags [-1,1)): -11~-2, -1~8
3	3a	429496730~483183820	53687091	[0,107374182)	Levels 429496730~483183819 (flags [0,107374180)): 9~1073741808 Level 483183820 (flags [107374180,107374182)): 1073741809~1073741824, 1073741825~1073741828
	3b	483183821~536870911	53687091	[107374182,214748364)	Levels 483183821~536870910 (flags [107374182,214748362)): 1073741829~2147383628 Level 536870911 (flags [214748362,214748364)): 2147483629~2147483646, 2147483647~-2147483648
	3c	536870912~590558003	53687092	[-214748364,-107374180)	Levels 536870912~590558002 (flags [-214748364,-107374182)): -2147483647~-1073741828 Level 590558003 (flags [-107374182,-107374180)): -1073741827~-1073741824, -1073741823~-1073741808
	3d	590558004~644245094	53687091	[-107374180,2)	Levels 590558004~644245093 (flags [-107374180,0)): -1073741807~-8 Level 644245094 (flags [0,2)): -7~-2, -1~12
4	4a	644245095~697932185	53687091	[1,107374183)	Levels 644245095~697932184 (flags [1,107374181)): 13~1073741812 Level 697932185 (flags [107374181,107374183)): 1073741813~1073741824, 1073741825~1073741832
	4b	697932186~751619276	53687091	[107374183,214748365)	Levels 697932186~751619275 (flags [107374183,214748363)): 1073741833~2147483632 Level 751619276 (flags [214748363,214748365)): 2147483633~2147483646, 2147483647~-2147483644
	4c	751619277~805306367	53687091	[-214748364,-107374182)	Levels 751619277~805306367 (flags [-214748364,-107374182)): -2147483643~-1073741824
	4d	805306368~858993459	53687092	[-107374182,2)	Levels 805306368~858993458 (flags [-107374182,0)): -1073741823~-4 Level 858993459 (flags [0,2)): -3~-2, -1~16
5	5a	858993460~912680550	53687091	[1,107374183)	Levels 858993460~912680549 (flags [1,107374181)): 17~1073741816 Level 912680550 (flags [107374181,107374183)): 1073741816~1073741824, 1073741825~1073741836
	5b	912680551~966367641	53687091	[107374183,214748365)	Levels 912680551~966367640 (flags [107374183,214748363)): 1073741837~2147483636 Level 966367641 (flags [214748363,214748365)): 2147483637~2147483646, 2147483647~-2147483640
	5c	966367642~1020054732	53687091	[-214748364,-107374182)	Levels 966367642~1020054731 (flags [-214748364,-107374184)): -2147483639~-1073741840 Level 1020054732 (flags [-107374184,-107374182)): -1073741839~-1073741824, -1073741823~-1073741820
	5d	1020054733~1073741823	53687091	[-107374182,0)	Levels 1020054733~1073741822 (flags [-107374182,-2)): -1073741819~-20 Level 1073741823 (flags [-2,0)): -19~-2, -1~0

List of telephone number ranges

I: ¹³⁴Ba ¹³⁵Ba ¹³⁶Ba ¹³⁷Ba ¹³⁸Ba ¹³⁹La ¹⁴⁷Sm ¹⁵⁰Sm ¹⁵¹Eu ¹⁵²Sm ¹⁵⁷Gd ¹⁵⁸Gd ¹⁵⁹Tb ¹⁷²Yb ¹⁷⁸Hf ¹⁸²W ¹⁸³W ¹⁸⁴W ¹⁸⁷Os ¹⁸⁸Os ¹⁹⁵Pt ¹⁹⁷Au ¹⁹⁸Hg

II: ¹³⁰Xe ¹³¹Xe ¹³²Xe ¹⁵⁵Gd ¹⁵⁶Gd ¹⁶⁶Er ¹⁷⁵Lu ¹⁷⁶Hf ¹⁸⁵Re ¹⁸⁶Os ¹⁹⁶Pt

III: ¹³³Cs ¹⁵³Eu ¹⁷³Yb ¹⁷⁷Hf ¹⁸⁰Hf ¹⁸¹Ta ¹⁸⁹Os ¹⁹⁰Os ¹⁹¹Ir ¹⁹³Ir ¹⁹⁹Hg

IV: ¹⁹²Pt

Virtual: ¹⁶²Dy ¹⁶⁵Ho ¹⁶⁷Er ¹⁷⁰Yb ¹⁷¹Yb

Factoring rational primes on the quadratic number field with discriminant $D$

• D = -376. Decomposing: A191056; remaining inert: A191086.
• D = -344. Decomposing: A191051; remaining inert: A191083.
• D = -312. Decomposing: A191047; remaining inert: A191080.
• D = -280. Decomposing: A191043; remaining inert: A191078.
• D = -248. Decomposing: A191040; remaining inert: A191076.
• D = -184. Decomposing: A191032; remaining inert: A191071.
• D = -163. Decomposing: A296921; remaining inert: A296915; not remaining inert: A257362.
• D = -152. Decomposing: A191028; remaining inert: A191069.
• D = -120. Decomposing: A191023; remaining inert: A191066.
• D = -95. Decomposing: A191057; remaining inert: A191087.
• D = -91. Decomposing: A191054; remaining inert: A191085.
• D = -88. Decomposing: A191020; remaining inert: A191064.
• D = -87. Decomposing: A191052; remaining inert: A191084.
• D = -83. Decomposing: A191050; remaining inert: A191082.
• D = -79. Decomposing: A191048; remaining inert: A191081.
• D = -71. Decomposing: A191044; remaining inert: A191079.
• D = -68. Decomposing: A296929; remaining inert: A296930; not remaining inert: A296931.
• D = -67. Decomposing: A191041; remaining inert: A191077; not remaining inert: A106933.
• D = -59. Decomposing: A191038; remaining inert: A191075.
• D = -56. Decomposing: A191017; remaining inert: A191061; not decomposing: A274504.
• D = -55. Decomposing: A191036; remaining inert: A191074.
• D = -52. Decomposing: A296926; remaining inert: A296927; not remaining inert: A296928 U {2}.
• D = -51. Decomposing: A191034; remaining inert: A191073.
• D = -47. Decomposing: A191033; remaining inert: A191072.
• D = -43. Decomposing: A191031; remaining inert: A184902; not remaining inert: A106891.
• D = -40. Decomposing: A155488; remaining inert: A296925; not remaining inert: A293859.
• D = -39. Decomposing: A191029; remaining inert: A191070.
• D = -35. Decomposing: A191026; remaining inert: A191068.
• D = -31. Decomposing: A191024; remaining inert: A191067.
• D = -24. Decomposing: A157437; remaining inert: A191059; not remaining inert: A296924.
• D = -23. Decomposing: A191021; remaining inert: A191065; not remaining inert: A296932.
• D = -20. Decomposing: A139513; remaining inert: A003626; not remaining inert: A240920 = A296922 U {2}; not decomposing: A296923 U {5}.
• D = -19. Decomposing: A191019; remaining inert: A191063; not remaining inert: A106863.
• D = -15. Decomposing: A191018; remaining inert: A191062.
• D = -11. Decomposing: A296920; remaining inert: A191060; not remaining inert: A056874.
• D = -8. Decomposing: A033200; remaining inert: A003628; not remaining inert: A033203; not decomposing: A045355.
• D = -7. Decomposing: A045386; remaining inert: A003625; not remaining inert: A045373; not decomposing: A045399.
• D = -4. Decomposing: A002144; remaining inert: A002145; not remaining inert: A002313; not decomposing: A045326.
• D = -3. Decomposing: A002476; remaining inert: A003627 = A007528 U {2}; not remaining inert: A007645; not decomposing: A045309 = A045410 U {2}.
• D = 5. Decomposing: A045468 = A064739 \ {2}; remaining inert: A003631 = A097957 U {2}; not remaining inert: A038872; not decomposing: A042993.
• D = 8. Decomposing: A001132 = A097958 \ {3}; remaining inert: A003629; not remaining inert: A038873; not decomposing: A042999.
• D = 12. Decomposing: A097933; remaining inert: A003630; not remaining inert: A038874 = A296933 U {2}; not decomposing: A038875 U {3}.
• D = 13. Decomposing: A296937; remaining inert: A038884; not remaining inert: A038883; not decomposing: A120330.
• D = 17. Decomposing: A296938; remaining inert: A038890; not remaining inert: A038889.
• D = 21. Remaining inert: A038894; not remaining inert: A038893.
• D = 24. Decomposing: A097934; remaining inert: A038877; not remaining inert: A038876.
• D = 28. Decomposing: A296934; remaining inert: A003632; not remaining inert: A038878.
• D = 29. Decomposing: A191022; remaining inert: A038902; not remaining inert: A038901.
• D = 33. Remaining inert: A038908; not remaining inert: A038907.
• D = 37. Decomposing: A191027; remaining inert: A038914; not remaining inert: A038913.
• D = 40. Decomposing: A097955; remaining inert: A038880; not remaining inert: A038879.
• D = 41. Decomposing: A191030; remaining inert: A038920; not remaining inert: A038919.
• D = 44. Decomposing: A296935; remaining inert: A296936; not remaining inert: A038881 U {2}; not decomposing: A038882 U {11}.
• D = 53. Decomposing: A191035; remaining inert: A038932; not remaining inert: A038931.
• D = 56. Remaining inert: A038886; not remaining inert: A038885.
• D = 57. Remaining inert: A038936; not remaining inert: A038935.
• D = 60. Decomposing: A097956; remaining inert: A038888; not remaining inert: A038887.
• D = 61. Decomposing: A191039; remaining inert: A038942; not remaining inert: A038941.
• D = 65. Remaining inert: A038946; not remaining inert: A038945.
• D = 69. Decomposing: A191042; remaining inert: A038952; not remaining inert: A038951.
• D = 73. Decomposing: A191045; remaining inert: A038958; not remaining inert: A038957.
• D = 76. Decomposing: A297175; remaining inert: A297176 = A038892 \ {2}; not remaining inert: A038891 U {2}.
• D = 77. Remaining inert: A038962; not remaining inert: A038961.
• D = 85. Remaining inert: A038972; not remaining inert: A038971.
• D = 88. Remaining inert: A038896; not remaining inert: A038895.
• D = 89. Decomposing: A191053; remaining inert: A038978; not remaining inert: A038977.
• D = 92. Decomposing: A297177; remaining inert: A038898; not remaining inert: A038897.
• D = 93. Decomposing: A191055; remaining inert: A038982; not remaining inert: A038981.
• D = 97. Decomposing: A191058; remaining inert: A038988; not remaining inert: A038987.
• D = 104. Remaining inert: A038900; not remaining inert: A038899.
• D = 120. Decomposing: A097959; remaining inert: A038904; not remaining inert: A038903.
• D = 124. Remaining inert: A038906; not remaining inert: A038905.
• D = 136. Decomposing: A191025; remaining inert: A038910; not remaining inert: A038909.
• D = 140. Remaining inert: A038912 \ {2}; not remaining inert: A038911 U {2}.
• D = 152. Remaining inert: A038916; not remaining inert: A038915.
• D = 156. Remaining inert: A038918; not remaining inert: A038917.
• D = 168. Remaining inert: A038922; not remaining inert: A038921.
• D = 172. Remaining inert: A038924 \ {2}; not remaining inert: A038923 U {2}.
• D = 184. Remaining inert: A038926; not remaining inert: A038925.
• D = 188. Remaining inert: A038928; not remaining inert: A038927.
• D = 204. Remaining inert: A038930 \ {2}; not remaining inert: A038929 U {2}.
• D = 220. Remaining inert: A038934; not remaining inert: A038933.
• D = 232. Decomposing: A191037; remaining inert: A038938; not remaining inert: A038937.
• D = 236. Remaining inert: A038940 \ {2}; not remaining inert: A038939 U {2}.
• D = 248. Remaining inert: A038944; not remaining inert: A038943.
• D = 264. Remaining inert: A038948; not remaining inert: A038947.
• D = 268. Remaining inert: A038950 \ {2}; not remaining inert: A038949 U {2}.
• D = 280. Remaining inert: A038954; not remaining inert: A038953.
• D = 284. Remaining inert: A038956; not remaining inert: A038955.
• D = 296. Decomposing: A191046; remaining inert: A038960; not remaining inert: A038959.
• D = 312. Remaining inert: A038964; not remaining inert: A038963.
• D = 316. Remaining inert: A038966; not remaining inert: A038965.
• D = 328. Decomposing: A191049; remaining inert: A038968; not remaining inert: A038967.
• D = 332. Remaining inert: A038970 \ {2}; not remaining inert: A038969 U {2}.
• D = 344. Remaining inert: A038974; not remaining inert: A038973.
• D = 348. Remaining inert: A038976; not remaining inert: A038975 U {2}.
• D = 364. Remaining inert: A038980 \ {2}; not remaining inert: A038979 U {2}.
• D = 376. Remaining inert: A038984; not remaining inert: A038983.
• D = 380. Remaining inert: A038986; not remaining inert: A038985.