Displaying 1-10 of 22 results found.
Largest metadrome (number with digits in strict ascending order) in base n.
+10
21
0, 1, 5, 27, 194, 1865, 22875, 342391, 6053444, 123456789, 2853116705, 73686780563, 2103299351334, 65751519677857, 2234152501943159, 81985529216486895, 3231407272993502984, 136146740744970718253, 6106233505124424657789, 290464265927977839335179
FORMULA
a(n) = Sum_{j=1...n-1} j*n^(n-1-j).
lim_{n->infinity} a(n)/a(n-1) - a(n-1)/a(n-2) = exp(1). - Conjectured by Gerald McGarvey, Sep 26 2004. Follows from the formula below and lim_{n->infinity} (1+1/n)^n = e. - Franklin T. Adams-Watters, Jan 25 2010
EXAMPLE
a(5) = 1234[5] (in base 5) = 1*5^3 + 2*5^2 + 3*5 + 4 = 125 + 50 + 15 + 4 = 194.
a(10) = 123456789 (in base 10).
MAPLE
0, seq((n^n-n^2+n-1)/(n-1)^2, n=2..100); # Robert Israel, Dec 13 2015
MATHEMATICA
Table[Total[(#1 n^#2) & @@@ Transpose@ {Range[n - 1], Reverse@ (Range[n - 1] - 1)}], {n, 20}] (* Michael De Vlieger, Jul 24 2015 *) Table[Sum[(b - k)*b^(k - 1), {k, b - 1}], {b, 30}] (* Clark Kimberling, Aug 22 2015 *) Table[FromDigits[Range[0, n - 1], n], {n, 20}] (* L. Edson Jeffery, Dec 13 2015 *)
PROG
(PARI) {for(i=1, 18, cuo=0; for(j=1, i-1, cuo=cuo+j*i^(i-j-1)); print1(cuo, ", "))} \\\ Douglas Latimer, May 16 2012 (Magma) [0] cat [(n^n-n^2+n-1)/(n-1)^2: n in [2..20]]; // Vincenzo Librandi, May 22 2012 (Haskell)
a023811 n = foldl (\val dig -> val * n + dig) 0 [0 .. n - 1]
(Python)
def a(n): return (n**n - n**2 + n - 1)//((n - 1)**2) if n > 1 else 0
Array read by ascending antidiagonals: T(n,k) is the k-th positive integer that is digitally balanced in base n.
+10
14
2, 11, 9, 75, 15, 10, 694, 78, 19, 12, 8345, 698, 99, 21, 35, 123717, 8350, 714, 108, 260, 37, 2177399, 123723, 8375, 722, 114, 266, 38, 44317196, 2177406, 123759, 8385, 738, 120, 268, 41, 1023456789, 44317204, 2177455, 123771, 8410, 742, 135, 278, 42
COMMENTS
A digitally balanced number in base b contains every digit from 0 to b-1 in equal amount.
EXAMPLE
Array begins:
n\k| 1 2 3 4 5 ...
-------------------------------------------------------------------------
2 | 2, 9, 10, 12, 35, ... = A031443 3 | 11, 15, 19, 21, 260, ... = A049354 4 | 75, 78, 99, 108, 114, ... = A049355 5 | 694, 698, 714, 722, 738, ... = A049356 6 | 8345, 8350, 8375, 8385, 8410, ... = A049357 7 | 123717, 123723, 123759, 123771, 123807, ... = A049358 8 | 2177399, 2177406, 2177455, 2177469, 2177518, ... = A049359 9 | 44317196, 44317204, 44317268, 44317284, 44317348, ... = A049360 10 | 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, ...
11 | 26432593615, 26432593625, 26432593725, 26432593745, 26432593845, ...
... | \______ A378001 (main diagonal) T(2,4) = 12 = 1100_2 is the fourth number in base 2 containing an equal amount of zeros and ones.
T(9,5) = 44317348 = 102345867_9 is the fifth number in base 9 containing an equal amount of digits from 0 to 8.
MATHEMATICA
Module[{dmax = 10, a, m}, a = Table[m = FromDigits[Join[{1, 0}, Range[2, n-1]], n] - 1; Table[While[!SameQ@@DigitCount[++m, n]]; m, dmax-n+2], {n, dmax+1, 2, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]
Numbers that have one of every digit in some base.
+10
11
2, 11, 15, 19, 21, 75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, 1054, 1070, 1102, 1110, 1138, 1142, 1178, 1190, 1202, 1210, 1294, 1298, 1334
COMMENTS
Also known as pandigital numbers, especially in base 10.
EXAMPLE
Base 3 values are 102_3 = 11, 120_3 = 15, 201_3 = 19, 210_3 = 21.
Triangle begins:
2;
11, 15, 19, 21;
75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, ...
694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, ...
...
MATHEMATICA
dtn[ L_, base_ ] := Fold[ base*#1+#2&, 0, L ] f[ n_ ] := Map[ dtn[ #, n ]&, Select[ Permutations[ Range[ 0, n-1 ] ], First[ # ]>0& ] ] Flatten[ Join[ Table[ f[ i ], {i, 2, 5} ] ] ]
CROSSREFS
Column k=1 gives A049363 (for n>1). Last elements of rows give A062813.
Number of pandigital squares (containing each digit exactly once) in base n.
+10
7
0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562
COMMENTS
For n = 18, the smallest and largest pandigital squares are 2200667320658951859841 and 39207739576969100808801. For n = 19, they are 104753558229986901966129 and 1972312183619434816475625. For n = 20, they are 5272187100814113874556176 and 104566626183621314286288961. - Chai Wah Wu, May 20 2015 When n is even, (n-1) is a factor of the pandigital squares. When n is odd, (n-1)/2 is a factor with the remaining factors being odd. Therefore, when n is odd and (n-1)/2 has an odd number of 2s as prime factors there are no pandigital squares in base n (e.g. 5, 13, 17 and 21). - Adam J.T. Partridge, May 21 2015 If n is odd and (n-1)/2 has an odd 2-adic valuation, then there are no squares in base n using all the digits from 1 to n-1 once, or all the digits from 0 to n-2 once or all the digits from 1 to n-2 once. This can be proved using the same argument as in the linked blogposts. - Chai Wah Wu, Feb 25 2024
EXAMPLE
For n=4 there is one pandigital square, 3201_4 = 225 = 15^2.
For n=6 there is one pandigital square, 452013_6 = 38025 = 195^2.
For n=10 there are 87 pandigital squares ( A036745). There are no pandigital squares in bases 2, 3, 5 or 13.
Hexadecimal has 3402 pandigital squares, the largest is FED5B39A42706C81.
PROG
(Python)
from gmpy2 import isqrt, mpz, digits
def A258103(n): # requires 2 <= n <= 62 c, sm, sq = 0, mpz(''.join([digits(i, n) for i in range(n-1, -1, -1)]), n), mpz(''.join(['1', '0']+[digits(i, n) for i in range(2, n)]), n)
m = isqrt(sq)
sq = m*m
m = 2*m+1
while sq <= sm:
if len(set(digits(sq, n))) == n:
c += 1
sq += m
m += 2
(PARI) a(n) = if(n%2==1 && valuation(n-1, 2)%2==0, 0, my(lim=sqrtint(n^n - (n^n-n)/(n-1)^2), count=0); for(m=sqrtint((n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1), lim, if(#Set(digits(m^2, n))==n, count++)); count) \\ Jianing Song, Feb 23 2024. Note that the searching range for m is [sqrt( A049363(n)), sqrt( A062813(n))]
Smallest square that is pandigital in base n.
+10
6
4, 64, 225, 5329, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, 307197306432025, 803752551280900, 29501156485626049, 1163446635475467225, 830482914641378019961, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176
COMMENTS
Compare this sequence with A260117, Smallest triangular number that is pandigital in base n. Presumably, lim_{n->infinity} A260117(n)/ A049363(n) = 1, but the same cannot be true for this sequence: the sum of the base-n digits of a number that is pandigital in base n must be 0+1+2+...+n-1 = binomial(n,2), but there are certain values of n for which no n-digit square can have a digit sum of binomial(n,2); for such values of n, a(n) must have more than n digits in base n. [E.g., the base-13 expansion of every square has a digit sum s == {0,1,4,9} (mod 12) (cf. A096008), but a square that is pandigital in base 13 and has exactly 13 digits would have a digit sum s = 78 == 6 (mod 12), so no such number exists; a 14-digit base-13 pandigital square would have each of the digits 0..12 exactly once except for one duplicated digit, which would have to be 3, 6, 7, or 10 (to yield a digit sum of 81, 84, 85, or 88, whose residues modulo 12 are 9, 0, 1, and 4, respectively). - Jon E. Schoenfield, Mar 23 2019] The values of n for which there exists no pandigital square that is exactly n digits long (in base n) begin with 2, 3, 5, 13, 17, 21, ...; presumably, for all such values of n, a(n) is exactly n+1 base-n digits long.
In base 2, there are no 2-digit squares at all, so a(2) must have more than 2 binary digits. For n = 3, 5, 13, 17, 21, ..., there exists no square, regardless of its number of digits, whose base-n digit sum equals binomial(n,2); see A260191.
EXAMPLE
Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms begin as follows:
.
n a(n) in base 10 a(n) in base n
== ========================= ======================
2 4 100_2
3 64 2101_3
4 225 3201_4
5 5329 132304_5
6 38025 452013_6
7 314721 2450361_7
8 3111696 13675420_8
9 61058596 136802574_9
10 1026753849 1026753849_10
11 31529329225 1240a536789_11
12 892067027049 124a7b538609_12
13 307197306432025 10254773ca86b9_13
14 803752551280900 10269b8c57d3a4_14
15 29501156485626049 102597bace836d4_15
16 1163446635475467225 1025648cfea37bd9_16
17 830482914641378019961 101246a89cgfb357ed_17
18 2200667320658951859841 10236b5f8eg4ad9ch7_18
19 104753558229986901966129 10234dhbg7ci8f6a9e5_19
20 5272187100814113874556176 1024e7cdi3hb695fja8g_20
a(n) is the smallest exponent such that prime(n)^k is pandigital in base 10.
+10
5
68, 39, 19, 18, 23, 22, 14, 17, 14, 12, 11, 13, 11, 13, 12, 13, 11, 14, 10, 15, 14, 13, 9, 11, 13, 9, 15, 14, 13, 12, 11, 15, 10, 7, 12, 9, 12, 10, 11, 8, 11, 8, 12, 11, 13, 13, 10, 12, 10, 8, 11, 12, 9, 7, 6, 7, 8, 12, 8, 8, 7, 7, 10, 9, 9, 6, 9, 10, 9, 10
COMMENTS
A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once.
REFERENCES
E.I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig-Jena-Berlin, 2. Auflage 1982.
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983.
EXAMPLE
2^68 = 295147905179352825856 (21 digits), 3^39 = 4052555153018976267 (19)
5^19 = 19073486328125 (14), 7^18 = 1628413597910449 (16), 11^23 = 895430243255237372246531 (24)
13^22 = 3211838877954855105157369 (25), 17^14 = 168377826559400929 (18)
19^17 = 5480386857784802185939 (22), 23^14 = 11592836324538749809 (20)
29^12 = 353814783205469041 (18), 31^11 = 25408476896404831 (17)
37^13 = 243569224216081305397 (21), 41^11 = 550329031716248441 (18)
43^13 = 1718264124282290785243 (22), 47^12 = 116191483108948578241 (21)
53^13 = 26036721925606486195973 (23), 59^11 = 30155888444737842659 (20)
61^14 = 9876832533361318095112441 (25), 67^10 = 1822837804551761449 (19)
71^15 = 5873205959385493353867330551 (28), 73^14 = 122045014039746588673695409 (23)
79^13 = 4668229371502258117133839 (25), 83^9 = 186940255267540403 (18)
89^11 = 2775173073766990340489 (22), 97^13 = 67302709016557486028618977 (26)
101^9 = 1093685272684360901 (19), 103^15 = 1557967416600764580522382952407 (31)
107^14 = 25785341502012466393542552649 (29), 109^13 = 306580461214335498944273629 (27)
113^12 = 4334523100191686738306881 (25), 127^11 = 138624799340320978519423 (24)
MATHEMATICA
sepan[n_]:=Module[{p=Prime[n], k=1}, While[Min[DigitCount[p^k]]==0, k++]; k]; Array[sepan, 100] (* Harvey P. Dale, Aug 03 2019 *)
PROG
(PARI) a(n) = {my(k=1, p=prime(n)); while(#Set(digits(p^k))<10, k++); k; } \\ Jinyuan Wang, Aug 14 2020
AUTHOR
Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 17 2009
Smallest triangular number that is pandigital in base n.
+10
5
6, 15, 78, 990, 8385, 128271, 2293011, 46923828, 1062489753, 27403863105, 757016521030, 24028339652778, 807863408487460, 29499468896141965, 1162871296355724735, 49093065731151773880, 2200689210818047715703, 104755000778178115071015, 5271254575974180914006953
COMMENTS
Presumably, lim_{n->infinity} a(n)/ A049363(n) = 1.
EXAMPLE
Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms are as follows:
.
n a(n) in base 10 a(n) in base n
== ========================= ==============
2 6 110_2
3 15 120_3
4 78 1032_4
5 990 12430_5
6 8385 102453_6
7 128271 1042653_7
8 2293011 10576423_8
9 46923828 107258346_9
10 1062489753 1062489753_10
11 27403863105 10692847a53_11
12 757016521030 10286b37459a_12
13 24028339652778 1053b2a49c786_13
14 807863408487460 1036cb2487d59a_14
15 29499468896141965 102568d3be749ca_15
16 1162871296355724735 102359486ac7edbf_16
17 49093065731151773880 1029a46d78g53cbef_17
18 2200689210818047715703 10237c486geh5bdaf9_18
19 104755000778178115071015 10236a47589cgdfeibh_19
20 5271254575974180914006953 10235i96e4jb8gfcah7d_20
PROG
(PARI) A049363(n)=if(n>1, (n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1, 1)
Greatest number of distinct digits of n in any base b>=2.
+10
4
1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3
COMMENTS
First occurrence of k: 1, 2, 11, 75, 694, 8345, 123717, ..., . A049363Last occurrence of k: 1, 43, 2462, 140081, 20338085, ..., . A246535Count of f(j) = k: 1, 22, 340, 8888, ..., . not in the OEIS data base. (End)
MATHEMATICA
f[n_] := Max@ Table[ Length@ Union@ IntegerDigits[n, b], {b, 2, n }]; f[1] = 1; Array[f, 105] (* Robert G. Wilson v, Aug 29 2014 *)
PROG
(PARI) a(n) = {nb = 1; for (k=2, n, nb = max (nb, #Set(digits(n, k))); ); nb; } \\ Michel Marcus, Sep 13 2014
CROSSREFS
See A037914 for the least corresponding bases.
Integers in which no more than half the digits (rounded up) are the same, for all bases up to ten.
+10
4
1, 2, 17, 19, 25, 38, 52, 56, 75, 76, 82, 83, 90, 92, 97, 98, 100, 102, 104, 105, 108, 113, 116, 135, 139, 141, 142, 147, 150, 153, 163, 165, 177, 178, 180, 184, 195, 197, 198, 201, 204, 209, 210, 212, 225, 226, 232, 267, 269, 275, 278, 279, 291, 293, 294, 298
EXAMPLE
267 in bases [2, 10] is 100001011, 100220, 10023, 2032, 1123, 531, 413, 326, 267. There are five zeros out of nine digits in its binary representation and no more than half the digits in the other bases are identical.
CROSSREFS
Cf. A049354, A049355, A049356, A049357, A049358, A049359, A049360, A049361, A049362, A049363, A049364
Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.
+10
3
2, 11, 8, 5, 75, 15, 7, 694, 119, 34, 8345, 1505, 195, 123717, 105803, 2217, 2134, 727, 2177399, 241934, 37303, 3724, 44317196, 4431858, 487068, 54771, 9124, 1023456789, 93125079, 9222117, 1110789, 92115, 26432593615
COMMENTS
For row n, the initial number is A049363(n) and the trailing number is A277056(n). Row 10 is A277054(1)- A277054(5) [note that the initial row is row 2].
EXAMPLE
The first rows of the array are:
2, (2, 2...)
11, 8, 5, (5, 5...)
75, 15, 7, (7, 7...)
694, 119, 34,
8345, 1505, 195,
123717, 105803, 2217, 2134, 727,
2177399, 241934, 37303, 3724,
44317196, 4431858, 487068, 54771, 9124,
1023456789, 93125079, 9222117, 1110789, 92115
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