A005900 -id:A005900 - cp's OEIS frontend

A053012 Platonic numbers: a(n) is a tetrahedral (A000292), cube (A000578), octahedral (A005900), dodecahedral (A006566) or icosahedral (A006564) number.

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

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 22 2000

19, the 3rd octahedral number, is the only prime platonic number. - Jean-François Alcover, Oct 11 2012

T. D. Noe, Table of n, a(n) for n = 1..1000
OEIS Wiki, Platonic numbers

Numbers of partitions into Platonic numbers: A226748, A226749.

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)
-- Reinhard Zumkeller, Jun 17 2013

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 *)

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*(3*n-1)*(3*n-2)/2, n*(5*n^2-5*n+2)/2) \\ Charles R Greathouse IV, Oct 11 2016

A175577 Decimal expansion of the sum of the reciprocals of the octahedral numbers (A005900).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jul 15 2010

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.

			1.2781851590909461795403909483...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A005900 (octahedral numbers).

Cf. sums of inverses: A152623 (tetrahedral numbers), A002117 (cubes), A295421 (dodecahedral numbers), A175578 (icosahedral numbers).

Digits := 120 : 3*(gamma+Psi(I/sqrt(2))); evalf(Re(%)) ;

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 *)

3*Euler+3*real(psi(I/sqrt(2))) \\ Charles R Greathouse IV, Jul 19 2013

sumnumrat(3/n/(2*n^2 + 1),1) \\ Charles R Greathouse IV, Feb 08 2023

A053678 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.

Original entry on oeis.org

5, 17, 191, 1412, 2143, 8393, 5346, 32475, 99234, 158712, 393981, 401023, 514617, 921485, 577350, 910495, 1430793, 3236406, 462236, 4122855

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000

			Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)

i values are A053676 and j values are A053677.

(* 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 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
a(13)-a(16) from Donovan Johnson, Jun 21 2010
a(17)-a(20) from Donovan Johnson, Sep 29 2010

A053676 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.

Original entry on oeis.org

7, 41, 465, 2732, 3005, 20648, 48125, 94396, 129299, 282931, 789281, 835050, 1241217, 1292143, 1521647, 1603655, 2756953, 4847702, 5128447, 6242598

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000

a(21) > 10^7. - Donovan Johnson, Sep 29 2010

			Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)

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.

Cf. A005900, A053677 (j values), A053678 (k values).

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
a(13)-a(16) from Donovan Johnson, Jun 21 2010
a(17)-a(20) from Donovan Johnson, Sep 29 2010

A053677 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.

Original entry on oeis.org

6, 40, 454, 2600, 2586, 20175, 48103, 93097, 105805, 265195, 755100, 803007, 1211000, 1111974, 1493421, 1499160, 2622000, 4309280, 5127195, 5574139

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 16 2000

			Oc(7) = 231 = Oc(6) + Oc(5); Oc(41) = 45961 = Oc(40) + Oc(17); Oc(465) = 67029905 = Oc(454) + Oc(191)

i values are A053676 and k values are A053678.

(* 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 *)

More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
a(13)-a(16) from Donovan Johnson, Jun 21 2010
a(17)-a(20) from Donovan Johnson, Sep 29 2010

A329072 Numbers that are sums of consecutive octahedral numbers (A005900).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 18 2019

nonn

Eric Weisstein's World of Mathematics, Octahedral Number

Cf. A005900, A006325, A217843, A322468, A322479, A329597, A329599.

A103981 Number of prime factors (with multiplicity) of octahedral numbers (A005900).

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 24 2005

When a(n) = 2, n is a term of A103982: indices of octahedral numbers (A005900) which are semiprimes.

			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.

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.

Robert Israel, Table of n, a(n) for n = 0..10000
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.
Eric Weisstein's World of Mathematics, Octahedral Number.

Cf. A001222, A005900, A103946, A103982.

seq(numtheory:-bigomega((2*n^3+n)/3),n=0..100); # Robert Israel, Aug 10 2014

a[n_] := PrimeOmega[n*(2*n^2 + 1)/3]; a[0] = 0; Array[a, 100, 0] (* Amiram Eldar, Oct 11 2024 *)

a(n) = A001222(A005900(n)), n>0. a(n) = Bigomega((2*n^3 + n)/3), n>0.

More terms from Wesley Ivan Hurt, Aug 11 2014

A103982 Indices of octahedral numbers (A005900) which are semiprimes.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Feb 23 2005

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.

			93 is in this sequence because A005900(93) = (2*93^3 + 93)/3 = 536269 = 31 * 17299, which is semiprime.

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2003), 65-75.
Jonathan Vos Post, Table of Polytope Numbers, Sorted, Through 1,000,000. [dead link]
Eric Weisstein's World of Mathematics, Octahedral Number.
Eric Weisstein's World of Mathematics, Semiprime.

Cf. A001222, A005900, A103946, A103981.

Flatten[Position[Table[(2n^3+n)/3,{n,1000}],?(PrimeOmega[#]==2&)]] (* _Harvey P. Dale, Jun 17 2013 *)

isok(n) = bigomega((2*n^3+n)/3) == 2; \\ Michel Marcus, Dec 14 2015

Numbers k such that A005900(k) is a term of A001358.
Numbers k such that A103981(k) = 2.
Numbers k such that A001222(A005900(k)) = 2.
Numbers k such that Bigomega((2*k^3 + k)/3) = 2.

More terms from Harvey P. Dale, Jun 17 2013

A133330 Sums of exactly three positive octahedral numbers A005900.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Oct 18 2007

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.

Agustin Moreno Canadas, On sums of figurate numbers by using techniques of poset representation theory, arXiv:0806.2486 [math.NT], 2008.

Cf. A005900, A053676, A053677, A053678.

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 *)

A131280 Sums of exactly 4 positive octahedral numbers A005900.

Original entry on oeis.org

4, 9, 14, 19, 22, 24, 27, 32, 37, 40, 45, 47, 50, 52, 57, 58, 62, 63, 65, 70, 75, 76, 83, 88, 90, 93, 95, 98, 100, 101, 103, 106, 108, 111, 113, 116, 124, 126, 129, 131, 133, 136, 138, 141, 142, 149, 151, 154, 159, 164, 167, 172, 174, 176, 177, 179, 182, 185, 190

Offset: 1

Jonathan Vos Post, Oct 21 2007

Pollock (1850) conjectured that every number is the sum of at most 7 octahedral numbers. Which octahedral numbers are themselves the sum of exactly 4 positive octahedral numbers? To begin with, Oc(3) = Oc(2) + Oc(2) + Oc(2) + Oc(1) = 6 + 6 + 6 + 1 = 19.

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.

Cf. A005900, A053676-A053678, A133330.

With[{octs=Table[(2n^3+n)/3,{n,10}]},Take[Union[Total/@Tuples[octs,4]], 60]] (* Harvey P. Dale, Nov 26 2013 *)

cp's OEIS Frontend

A053012 Platonic numbers: a(n) is a tetrahedral (A000292), cube (A000578), octahedral (A005900), dodecahedral (A006566) or icosahedral (A006564) number.

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A175577 Decimal expansion of the sum of the reciprocals of the octahedral numbers (A005900).

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A053678 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.

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A053676 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.

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A053677 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.

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A329072 Numbers that are sums of consecutive octahedral numbers (A005900).

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A103981 Number of prime factors (with multiplicity) of octahedral numbers (A005900).

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A103982 Indices of octahedral numbers (A005900) which are semiprimes.

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