Displaying 1-10 of 125 results found.
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1, 4, 6, 8, 10, 12, 19, 20, 27, 35, 44, 48, 56, 64, 84, 85, 120, 124, 125, 146, 165, 216, 220, 231, 255, 286, 343, 344, 364, 455, 456, 489, 512, 560, 670, 680, 729, 742, 816, 891, 969, 1000, 1128, 1140, 1156, 1330, 1331, 1469, 1540, 1629, 1728, 1771, 1834
COMMENTS
All dodecahedral numbers are tetrahedral, so equivalently numbers that are tetrahedral , cubic, octahedral, or icosahedral. - Charles R Greathouse IV, Sep 22 2025
FORMULA
a(n) ~ k*n^3 where k = 1/(6^(1/3) + (3/2)^(1/3) + (2/5)^(1/3) + 1)^3 = 0.00964... is the unique real root of a 27th-degree polynomial. - Charles R Greathouse IV, Sep 29 2025
MATHEMATICA
nn = 25; t1 = Table[n (n + 1) (n + 2)/6, {n, nn}]; t2 = Table[n^3, {n, nn}]; t3 = Table[(2*n^3 + n)/3, {n, nn}]; t4 = Table[n (3*n - 1) (3*n - 2)/2, {n, nn}]; t5 = Table[n (5*n^2 - 5*n + 2)/2, {n, nn}]; Select[Union[t1, t2, t3, t4, t5], # <= t1[[-1]] &] (* T. D. Noe, Oct 13 2012 *)
PROG
(Haskell)
a053012 n = a053012_list !! (n-1)
a053012_list = tail $ f
[a000292_list, a000578_list, a005900_list, a006566_list, a006564_list]
where f pss = m : f (map (dropWhile (<= m)) pss)
where m = minimum (map head pss)
(PARI) listpoly(lim, poly[..])=my(v=List()); for(i=1, #poly, my(P=poly[i], x=variable(P), f=k->subst(P, x, k), n, t); while((t=f(n++))<=lim, listput(v, t))); Set(v)
list(lim)=my(n='n); listpoly(lim, n*(n+1)*(n+2)/6, n^3, (2*n^3+n)/3, n*(5*n^2-5*n+2)/2) \\ Charles R Greathouse IV, Oct 11 2016, updated Sep 22 2025
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 22 2000
Decimal expansion of the sum of the reciprocals of the octahedral numbers ( A005900).
+20
7
1, 2, 7, 8, 1, 8, 5, 1, 5, 9, 0, 9, 0, 9, 4, 6, 1, 7, 9, 5, 4, 0, 3, 9, 0, 9, 4, 8, 3, 6, 7, 5, 7, 1, 3, 3, 8, 4, 2, 3, 9, 0, 1, 5, 3, 6, 8, 5, 1, 4, 0, 2, 0, 2, 0, 1, 7, 0, 3, 4, 6, 3, 8, 0, 4, 1, 6, 5, 7, 9, 9, 9, 1, 8, 3, 0, 6, 2, 0, 8, 2, 4, 4, 1, 8, 3, 6, 3, 2, 4, 5, 2, 0, 5, 0, 0, 7, 9, 6, 2, 3, 0, 5, 3, 9
COMMENTS
Defined by Sum_{n>=1} 1/ A005900(n) = 1/1 + 1/6 + 1/19 + 1/44 + ... Equals 3*(gamma + Re psi(i/sqrt 2) ) = 3* Re( A001620 + psi(i* A010503)) where psi(i* A010503) = -0.1511539... + i*2.3152942... is a digamma function and i the imaginary unit.
EXAMPLE
1.2781851590909461795403909483...
MAPLE
Digits := 120 : 3*(gamma+Psi(I/sqrt(2))); evalf(Re(%)) ;
MATHEMATICA
RealDigits[ 3/2*(2*EulerGamma + Re[PolyGamma[0, 1 - I/Sqrt[2]] + PolyGamma[0, 1 + I/Sqrt[2]]]), 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)
Let Oc(n) = A005900(n) = n-th octahedral number. Consider all integer triples (i,j,k), j >= k > 0, with Oc(i) = Oc(j)+Oc(k), ordered by increasing i; sequence gives k values.
+20
6
5, 17, 191, 1412, 2143, 8393, 5346, 32475, 99234, 158712, 393981, 401023, 514617, 921485, 577350, 910495, 1430793, 3236406, 462236, 4122855
EXAMPLE
Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)
MATHEMATICA
(* This is just a check of k-values, given i-values *) A053676 = {7, 41, 465, 2732, 3005, 20648, 48125, 94396, 129299, 282931, 789281, 835050, 1241217, 1292143, 1521647, 1603655, 2756953, 4847702, 5128447, 6242598}; r[i_] := Reduce[0 < k <= j && 2*i^3 + i == 2*j^3 + j + 2*k^3 + k, {j, k}, Integers]; A053678 = Table[res = {i, j, k} /. ToRules[r[i]]; Print[res]; res, {i, A053676}][[All, 3]] (* Jean-François Alcover, Dec 07 2012 *)
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
Let Oc(n) = A005900(n) = n-th octahedral number. Consider all integer triples (i,j,k), j >= k > 0, with Oc(i) = Oc(j)+Oc(k), ordered by increasing i; sequence gives i values.
+20
5
7, 41, 465, 2732, 3005, 20648, 48125, 94396, 129299, 282931, 789281, 835050, 1241217, 1292143, 1521647, 1603655, 2756953, 4847702, 5128447, 6242598
REFERENCES
Pollock, F. "On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders." Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
Dickson, L. E., History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005, cites the Pollock reference.
EXAMPLE
Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
Let Oc(n) = A005900(n) = n-th octahedral number. Consider all integer triples (i,j,k), j >= k > 0, with Oc(i) = Oc(j)+Oc(k), ordered by increasing i; sequence gives j values.
+20
5
6, 40, 454, 2600, 2586, 20175, 48103, 93097, 105805, 265195, 755100, 803007, 1211000, 1111974, 1493421, 1499160, 2622000, 4309280, 5127195, 5574139
EXAMPLE
Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)
MATHEMATICA
(* This is just a check of j-values, given i-values *) A053676 = {7, 41, 465, 2732, 3005, 20648, 48125, 94396, 129299, 282931, 789281, 835050, 1241217, 1292143, 1521647, 1603655, 2756953, 4847702, 5128447, 6242598}; r[i_] := Reduce[0 < k <= j && 2*i^3 + i == 2*j^3 + j + 2*k^3 + k, {j, k}, Integers]; A053677 = Table[res = {i, j, k} /. ToRules[r[i]]; Print[res]; res, {i, A053676}][[All, 2]] (* Jean-François Alcover, Dec 07 2012 *)
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
Numbers that are sums of one or more consecutive octahedral numbers ( A005900).
+20
4
0, 1, 6, 7, 19, 25, 26, 44, 63, 69, 70, 85, 129, 146, 148, 154, 155, 231, 275, 294, 300, 301, 344, 377, 462, 489, 506, 525, 531, 532, 575, 670, 721, 806, 833, 850, 869, 875, 876, 891, 1064, 1156, 1159, 1210, 1295, 1339, 1358, 1364, 1365, 1469, 1503, 1561, 1734
COMMENTS
The first term that is a sum of consecutive octahedral numbers in more than one way is a(18) = 231 = A005900(7) = 85 + 146 = A005900(5) + A005900(6). Are there any terms that are a sum of consecutive octahedral numbers in three ways? - Robert Israel, Jul 20 2026
MAPLE
N:= 10^4: # for terms <= N
Oct:= [seq(n*(2*n^2+1)/3, n=0..floor(fsolve(x*(2*x^2+1)/3=N)))]:
S:= ListTools:-PartialSums(Oct):
select(t -> t <= N, sort(convert({0, seq(seq(S[j]-S[i], i=1..j-1), j=1..nops(S))}, list))); # Robert Israel, Jul 20 2026
Number of prime factors (with multiplicity) of octahedral numbers ( A005900).
+20
2
0, 0, 2, 1, 3, 2, 2, 3, 4, 2, 3, 5, 4, 2, 3, 3, 7, 2, 4, 2, 5, 2, 4, 2, 4, 4, 4, 3, 4, 4, 3, 2, 6, 2, 4, 4, 4, 3, 5, 3, 6, 3, 3, 4, 4, 3, 4, 3, 6, 3, 4, 4, 5, 2, 5, 3, 7, 3, 3, 3, 5, 3, 4, 4, 7, 5, 3, 3, 4, 3, 8, 2, 5, 4, 4, 3, 4, 4, 4, 4, 7, 5, 3, 3, 5, 3, 3
COMMENTS
When a(n) = 2, n is a term of A103982: indices of octahedral numbers ( A005900) which are semiprimes.
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York, Springer-Verlag, p. 50, 1996.
L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.
EXAMPLE
a(3) = 1 because OctahedralNumber(3) = A005900(3) = 19, which is prime and thus has only one prime factor. Because the cubic polynomial for octahedral numbers factors into n time a quadratic, the octahedral numbers can never be prime after a(3) = 19. a(4) = 3 because A005900(4) = (2*4^3 + 4)/3 = 44 = 2 * 2 * 11, which has (with multiplicity) three prime factors.
MAPLE
seq(numtheory:-bigomega((2*n^3+n)/3), n=0..100); # Robert Israel, Aug 10 2014
MATHEMATICA
a[n_] := PrimeOmega[n*(2*n^2 + 1)/3]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Oct 11 2024 *)
Indices of octahedral numbers ( A005900) which are semiprimes.
+20
2
2, 5, 6, 9, 13, 17, 19, 21, 23, 31, 33, 53, 71, 87, 89, 93, 113, 123, 127, 157, 163, 167, 177, 181, 197, 201, 219, 229, 237, 321, 327, 347, 373, 393, 401, 409, 417, 419, 449, 487, 489, 503, 509, 519, 523, 537, 541, 563, 571, 577, 597, 599, 633, 647, 699, 751
COMMENTS
Because the cubic polynomial for octahedral numbers factors into n time a quadratic, the octahedral numbers can never be prime after a(3) = 19, but can be semiprime (if n is prime and 2*n^2+1 is triple a prime, or if n is triple a prime and 2*n^2+1 is prime). A005900(37) = 33781 = 11 * 37 * 83, three prime factors with same number of digits. A005900(41) = 45961 = 19 * 41 * 59, three prime factors with same number of digits. A005900(57) = 123481 = 19 * 67 * 97, three prime factors with same number of digits. A005900(67) = 200531 = 41 * 67 * 73, three prime factors with same number of digits. A005900(73) = 259369 = 11 * 17 * 19 * 73, four prime factors with same number of digits.
REFERENCES
J. H. Conway and R. K. Guy, The Book of Numbers, New York, Springer-Verlag, p. 50, 1996.
L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, 1952.
LINKS
Eric Weisstein's World of Mathematics, Semiprime.
FORMULA
Numbers k such that A103981(k) = 2. Numbers k such that Bigomega((2*k^3 + k)/3) = 2.
EXAMPLE
93 is in this sequence because A005900(93) = (2*93^3 + 93)/3 = 536269 = 31 * 17299, which is semiprime.
MATHEMATICA
Flatten[Position[Table[(2n^3+n)/3, {n, 1000}], _?(PrimeOmega[#]==2&)]] (* Harvey P. Dale, Jun 17 2013 *)
PROG
(PARI) isok(n) = bigomega((2*n^3+n)/3) == 2; \\ Michel Marcus, Dec 14 2015
Sums of exactly three positive octahedral numbers A005900.
+20
1
3, 8, 13, 18, 21, 26, 31, 39, 44, 46, 51, 56, 57, 64, 69, 82, 87, 89, 92, 94, 97, 105, 107, 110, 123, 130, 132, 135, 148, 153, 158, 166, 171, 173, 176, 184, 189, 191, 196, 209, 214, 232, 233, 234, 237, 238, 243, 250, 251, 255, 256, 269, 275, 276, 281, 293, 294
REFERENCES
Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005, cites the Pollock reference.
Pollock, F. "On the Extension of the Principle of Fermat's Theorem of the Polygonal Numbers to the Higher Orders of Series Whose Ultimate Differences Are Constant. With a New Theorem Proposed, Applicable to All the Orders." Abs. Papers Commun. Roy. Soc. London 5, 922-924, 1843-1850.
MATHEMATICA
lim = 300; oc[n_] := (2*n^3 + n)/3; nmax = Floor[Solve[oc[n] + oc[1] + oc[1] == lim, n][[1, 1, 2]]]; t = Table[ oc[n], {n, nmax}]; Select[ Union[ Flatten[ Outer[ Plus, t, t, t]]], # <= lim &] (* Jean-François Alcover, Sep 08 2011 *)
Decimal expansion of Product_{k>=1} (1 + 1/ A005900(k)) (octahedral numbers).
+20
1
2, 6, 0, 4, 1, 6, 7, 7, 5, 1, 3, 7, 9, 5, 2, 1, 3, 5, 3, 9, 4, 9, 9, 2, 1, 5, 3, 9, 1, 8, 3, 4, 2, 4, 1, 6, 7, 3, 9, 8, 1, 1, 6, 8, 7, 2, 8, 2, 2, 2, 2, 6, 0, 1, 1, 3, 5, 4, 6, 4, 3, 3, 5, 7, 4, 0, 2, 0, 5, 8, 6, 0, 3, 6, 5, 2, 3, 6, 9, 2, 2, 8, 0, 2, 0, 5, 4, 8, 1, 7, 3, 6, 1, 2, 5, 1, 8, 1, 2, 4, 7, 5, 6, 5, 1
FORMULA
Equals (cosh((sqrt(5)*Pi)/2)*csch(Pi/sqrt(2)))/sqrt(2).
MAPLE
c:= simplify(product(1+1/(n*(2*n^2+1)/3), n=1..infinity)):
MATHEMATICA
N[(Cosh[(Sqrt[5] Pi)/2] Csch[Pi/Sqrt[2]])/Sqrt[2], 79]
PROG
(PARI) prodnumrat(1 + 1/('k*(2*'k^2 + 1)/3), 1)
(PARI) cosh((sqrt(5)*Pi)/2)/sinh(Pi/sqrt(2))/sqrt(2)
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