A002311 -id:A002311 - cp's OEIS frontend

A133459 Sums of two nonzero pentagonal pyramidal numbers.

2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, 294, 306, 322, 328, 363, 392, 406, 411, 414, 423, 445, 480, 484, 531, 551, 556, 568, 576, 590, 601, 625, 676, 693, 727, 732, 744, 746, 766

Offset: 1

Jonathan Vos Post, Dec 23 2007

nonn

Does this sequence ever include a pentagonal pyramidal number? That is, is it ever the case that A002411(i)=A002411(j)+A002411(k) as is often true for triangular pyramidal numbers (tetrahedral numbers) or square pyramidal numbers?

The answer to the above question is yes: A002411(30) + A002411(36) = 13950 + 23976 = 37926 = A002411(42) (see A172425). - Chai Wah Wu, Apr 16 2016

Cf. A002311, A002411, A053721, A172425.

nn = 12; Take[Union@ Map[Total, Tuples[#^2 (# + 1)/2 &@ Range@ nn, 2]], # (# - 1)/2 &[nn - 1]] (* Michael De Vlieger, Apr 16 2016 *)

{A002411(i) + A002411(j) for i, j > 0} = {i^2*(i+1)/2 + j^2*(j+1)/2 for i, j > 0}.

A034404 Values of C(n,3) which can be written as C(x,3) + C(y,3).

Original entry on oeis.org

20, 680, 29260, 34220, 70300, 221815, 227920, 287980, 467180, 908600, 2481115, 4278680, 12259940, 13813570, 15493204, 17861900, 19970444, 24672560, 25665020, 27880600, 29742164, 34055980, 44722580

Offset: 1

N. J. A. Sloane

Bombieri's Napkin Problem: Bombieri said that "the equation C(x,n) + C(y,n) = C(z,n) has no trivial solutions for n >= 3" (the joke being that he said "trivial" rather than "nontrivial"!).

Also: tetrahedral numbers that are the sum of two other tetrahedral numbers. (For the indices of these terms, see A002311.) - Harvey P. Dale, Jul 25 2011

			C(10,3) + C(16,3) = C(17,3) = 680.

Van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 122.

Delbert L. Johnson, Table of n, a(n) for n = 1..672

Cf. A000292, A010330, A002311.

a034404 = a000292 . a002311  -- Reinhard Zumkeller, May 02 2014

With[{tetras=Binomial[Range[700]+2,3]},Union[Select[Total/@Tuples[ tetras,2], MemberQ[tetras,#]&]]] (* Harvey P. Dale, Jul 25 2011 *)

a(n) = A000292(A002311(n)). - Reinhard Zumkeller, May 02 2014

Offset corrected by Andrew Howroyd, Mar 23 2023

A010330 Numbers k such that C(k,3) = C(x,3) + C(y,3) is solvable.

Original entry on oeis.org

6, 17, 57, 60, 76, 111, 112, 121, 142, 177, 247, 296, 420, 437, 454, 476, 494, 530, 537, 552, 564, 590, 646, 690, 704, 716, 742, 749, 755, 820, 870, 910, 920, 1100, 1160, 1222, 1243, 1430, 1436, 1446, 1452, 1647, 1710, 1740, 1788, 1870, 2172, 2185, 2222, 2258

Offset: 1

N. J. A. Sloane

Bombieri's Napkin Problem: Bombieri said that "the equation C(x,n)+C(y,n)=C(z,n) has no trivial solutions for n >= 3" (the joke being that he said "trivial" rather than "nontrivial"!).

			C(10,3) + C(16,3) = C(17,3) = 680, so 17 is a term.

J. Leech, Some solutions of Diophantine equations, Proc. Camb. Phil. Soc., 53 (1957), 778-780.
Van der Poorten, Notes on Fermat's Last Theorem, Wiley, p. 122.

T. D. Noe, Table of n, a(n) for n = 1..463 (n < 10^6)

Cf. A034404.

Cf. A000292.

a010330 = (+ 2) . a002311  -- Reinhard Zumkeller, May 02 2014

f[n_]:=Reduce[1 < x <= y < n && n(n-1)(n-2) == x(x-1)(x-2) + y(y-1)(y-2), {x,y}, Integers]; Select[Range[2260], (f[#] =!= False)&] (* Jean-François Alcover, Mar 30 2011 *)

a(n) = A002311(n) + 2. - Reinhard Zumkeller, May 02 2014

More terms from David W. Wilson

A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

36, 81, 144, 289, 484, 576, 625, 676, 3600, 7396, 9801, 14400, 35344, 40000, 40804, 44100, 45796, 56644, 59049, 71824, 112896, 121104, 172225, 226576, 231361, 254016, 274576, 290521, 319225, 362404, 480249, 495616, 518400, 527076, 535824

Offset: 1

Alexander Adamchuk, Dec 25 2007

nonn

Corresponding numbers m such that m^2 = a(n) are listed in A136360.
Note that some numbers in A136360 are also perfect squares. The corresponding numbers k such that m = k^2 are listed in A136361.
Includes all nonzero members of A099764: this occurs when the two pentagonal pyramidal numbers are both equal to i^2*(i+1)/2 where i+1 is a square. - Robert Israel, Feb 04 2020

			A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.
Thus a(1) = 36, a(2) = 81, a(3) = 144, a(4) = 289 that are the perfect squares in A133459.

Robert Israel, Table of n, a(n) for n = 1..985

Cf. A136360, A136361, A133459, A002311, A002411, A053721, A099764.

N:= 200: # for terms up to N^2*(N+1)/2.
PP:= [seq(i^2*(i+1)/2, i=1..N)]:
PP2:= sort(convert(select(`<=`,{seq(seq(PP[i]+PP[j],j=i..N),i=1..N)},PP[-1]),list)):
select(issqr,PP2); # Robert Israel, Feb 04 2020

Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ]

a(n) = A136360(n)^2.

A136360 Square roots of the perfect squares in A133459.

Original entry on oeis.org

6, 9, 12, 17, 22, 24, 25, 26, 60, 86, 99, 120, 188, 200, 202, 210, 214, 238, 243, 268, 336, 348, 415, 476, 481, 504, 524, 539, 565, 602, 693, 704, 720, 726, 732, 846, 899, 961, 965, 990, 1026, 1202, 1218, 1221, 1224, 1320, 1551, 1602, 1687, 1716, 1724, 1734

Offset: 1

Alexander Adamchuk, Dec 25 2007

nonn

Corresponding squares in A133459 are listed in A136359(n) = a(n)^2.

Note that some numbers in a(n) are also perfect squares: m = k^2 = {9, 25, 961, 17424, ...}. The corresponding numbers k such that a(n) = k^2 are listed in A136361.

			A133459 begins {2, 7, 12, 19, 24, 36, 41, 46, 58, 76, 80, 81, 93, 115, 127, 132, 144, 150, 166, 197, 201, 202, 214, 236, 252, 271, 289, ...}.
Thus a(1) = sqrt(36) = 6, a(2) = sqrt(81) = 9, a(3) = sqrt(144) = 12, a(4) = sqrt(289) = 17 that are the square roots of the perfect squares in A133459.

Cf. A136359, A136361, A133459, A002311, A002411, A053721.

Sqrt[ Select[ Intersection[ Flatten[ Table[ i^2*(i+1)/2 + j^2*(j+1)/2, {i,1,300}, {j,1,i} ] ] ], IntegerQ[ Sqrt[ # ] ] & ] ]

a(n) = sqrt(A136359(n)).

A136361 Square roots of the perfect squares in A136360; or numbers k such that k^4 is in A133459 = the sums of two nonzero pentagonal pyramidal numbers.

Original entry on oeis.org

3, 5, 31, 132, 1068, 9672, 50664, 145060

Offset: 1

Alexander Adamchuk, Dec 25 2007

Corresponding perfect squares in A136360 are a(n)^2 = {9, 25, 961, 17424, ...}. They correspond to the perfect fourth powers in A133459 = Sums of two nonzero pentagonal pyramidal numbers. a(n)^4 are the terms of A133459: {81, 525, 923521, 303595776, ...}. Note that the first three terms are prime.

a(9) > (5*10^20)^(1/4). - Donovan Johnson, Jun 12 2011

Cf. A136359, A136360, A133459, A002311, A002411, A053721.

Name corrected and a(5)-a(6) from Donovan Johnson, Nov 20 2010

a(7)-a(8) from Donovan Johnson, Jun 12 2011

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A133459 Sums of two nonzero pentagonal pyramidal numbers.

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A034404 Values of C(n,3) which can be written as C(x,3) + C(y,3).

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A010330 Numbers k such that C(k,3) = C(x,3) + C(y,3) is solvable.

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A136359 Perfect squares in A133459; or perfect squares that are the sums of two nonzero pentagonal pyramidal numbers.

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A136360 Square roots of the perfect squares in A133459.

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A136361 Square roots of the perfect squares in A136360; or numbers k such that k^4 is in A133459 = the sums of two nonzero pentagonal pyramidal numbers.

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