A225908 -id:A225908 - cp's OEIS frontend

A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

6, 9, 12, 18, 19, 20, 24, 25, 27, 28, 29, 30, 36, 38, 40, 41, 42, 44, 45, 46, 48, 50, 53, 54, 56, 57, 58, 60, 63, 66, 67, 69, 70, 71, 72, 75, 76, 78, 80, 81, 82, 84, 85, 87, 88, 89, 90, 92, 93, 95, 96, 97, 99, 100, 102, 103, 105, 106, 108, 110, 111, 112, 113

Offset: 1

N. J. A. Sloane

nonn

Numbers w such that w^3 = x^3+y^3+z^3, x>y>z>=0, is soluble.
A226903(n) + 1 is an infinite subsequence parametrized by Shiraishi in 1826. - Jonathan Sondow, Jun 22 2013
Because of Fermat's Last Theorem, sequence lists numbers w such that w^3 = x^3+y^3+z^3, x>y>z>0, is soluble. In other words, z cannot be 0 because x and y are positive integers by definition of this sequence. - Altug Alkan, May 08 2016
This sequence is the same as numbers w such that w^3 = x^3+y^3+z^3, x>=y>=z>0, is soluble as Legendre showed that a^3+b^3=2*c^3 only has the trivial solutions a = b or a = -b (see Dickson's History of the Theory of Numbers, vol. II, p. 573). - Chai Wah Wu, May 13 2017

			20 belongs to the sequence as 20^3 = 7^3 + 14^3 + 17^3.

Ya. I. Perelman, Algebra can be fun, pp. 142-143.

Nathaniel Johnston and Chai Wah Wu, Table of n, a(n) for n = 1..10000 (1..1500 from Nathaniel Johnston)
A. Russell and C. E. Gwyther, The Partition of Cubes, The Mathematical Gazette, 21 (1937), pp. 33-35.
Index to sequences related to Diophantine equations (3,1,3)

Cf. A001235, A003072, A114923, A225908, A226903.

for w from 1 to 113 do for z from 0 to w-1 do bk:=0: for y from z+1 to w-1 do for x from y+((w+z) mod 2) to w-1 by 2 do if(x^3+y^3+z^3=w^3)then printf("%d, ",w); bk:=1: break: fi: od: if(bk=1)then break: fi: od: if(bk=1)then break: fi: od: od: # Nathaniel Johnston, Jun 22 2013

Select[Range[200], n |-> Length[PowersRepresentations[n^3, 3, 3]] > 1] (* Paul C Abbott, May 07 2025 *)

has(n)=my(L=sqrtnint(n-1,3)+1, U=sqrtnint(4*n,3)); fordiv(n,m, if(L<=m && m<=U, my(ell=(m^2-n/m)/3); if(denominator(ell)==1 && issquare(m^2-4*ell), return(1)))); 0
list(lim)=my(v=List(),a3,t); lim\=1; for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim, sqrtnint(a3-lim-1,3)+1,1), a-1, t=a3-b^3; if(has(t), listput(v,a)))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

A226903 Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.

Original entry on oeis.org

5, 18, 53, 102, 197, 306, 491, 684, 989, 1290, 1745, 2178, 2813, 3402, 4247, 5016, 6101, 7074, 8429, 9630, 11285, 12738, 14723, 16452, 18797, 20826, 23561, 25914, 29069, 31770, 35375, 38448, 42533, 46002, 50597, 54486, 59621, 63954, 69659, 74460, 80765, 86058

Offset: 1

Jonathan Sondow, Jun 22 2013

Shiraishi's solutions to a^3 + b^3 + c^3 = d^3 are a = 3n^2; b = 6n^2 - 3n + 1 or 6n^2 + 3n + 1; c = 9n^3 - 6n^2 + 3n - 1 or  9n^3 + 6n^2 + 3n, respectively, for n > 0; and d = c+1. See Smith and Mikami for a derivation.
Shiraishi's formulas show that the sequence is infinite. Hence the sequences A023042 (solutions to x^3 + y^3 + z^3 = w^3), A225908 (solutions to a^3 + b^3 = c^3 - d^3), A225909 (solutions to a^3 + b^3 = (c+1)^3 - c^3) and A226902 (numbers c in A225909) are also infinite.
Shiraishi's solution b = 6n^2 +/- 3n + 1 is the centered triangular numbers A005448 except 1.

			The first two terms are a(1) = 9 - 6 + 3 - 1 = 5 and a(2) = 9 + 6 + 3 = 18. Then Shiraishi's formulas give 3^3 + 4^3 + 5^3 = 6^3 and 3^3 + 10^3 + 18^3 = 19^3.

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
Wikipedia (French), Shiraishi Nagatada
Wikipedia (German), Shiraishi Nagatada
Index entries for sequences related to sums of cubes
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A003325, A005448, A023042, A181123, A225908, A225909, A226902.

a(2n-1) = 9n^3 - 6n^2 + 3n - 1.
a(2n) = 9n^3 + 6n^2 + 3n.
G.f.: x*(5 + 13*x + 20*x^2 + 10*x^3 + 5*x^4 + x^5) / ((1 + x)^3*(1 - x)^4). [Bruno Berselli, Jun 22 2013]
a(n) = (18*n^3 + 27*n^2 + 27*n + 1 - (3*n^2 + 3*n + 1)*(-1)^n)/16. [Bruno Berselli, Jun 22 2013]
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n > 7. - Chai Wah Wu, Aug 05 2025

A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.

Original entry on oeis.org

91, 217, 1027, 4921, 8587, 14911, 31519, 39331, 106597, 117019, 136747, 185257, 195841, 265519, 281827, 616987, 636181, 684019, 712969, 724717, 736561, 955981, 1200169, 1352737, 1405621, 1771777, 2481571, 2756167, 2937331, 4251871, 4996171, 5262901

Offset: 1

Jonathan Sondow, Jun 21 2013

nonn

Solutions x to the equations x = a^3 + b^3 = (c+1)^3 - c^3 in positive integers. The values of c are A226902.
The intersection of A003325 and A003215.
Subsequence of A225908 = numbers that are both a sum and a difference of two positive cubes.
Shiraishi's solution to Gokai Ampon's equation u^3 + v^3 + w^3 = n^3 (see A023042 and A226903) shows that the sequence is infinite.

			3^3 + 4^3 = 6^3 - 5^3 = 91, so 91 is a member.

Shiraishi Chochu (aka Shiraishi Nagatada), Shamei Sampu (Sacred Mathematics), 1826.

Chai Wah Wu, Table of n, a(n) for n = 1..5000
David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics, Open Court, Chicago, 1914; Dover reprint, 2004; pp. 233-235.
Wikipedia (French), Shiraishi Nagatada
Wikipedia (German), Shiraishi Nagatada
Index entries for sequences related to sums of cubes

Cf. A003215, A003325, A023042, A181123, A225908, A226902, A226903.

Select[#[[2]]-#[[1]]&/@Partition[Range[5000]^3,2,1],Count[ IntegerPartitions[ #,{2}],?(AllTrue[Surd[#,3],IntegerQ]&)]>0&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale, Sep 07 2018 *)

a(n) = (A226902(n)+1)^3 - A226902(n)^3.

A230717 Squares that are both a sum and a difference of two positive cubes.

Original entry on oeis.org

345744, 1058841, 1750329, 8340544, 22127616, 67765824, 68574961, 95004009, 112021056, 252047376, 533794816, 771895089, 1097199376, 1232922769, 1275989841, 1416167424, 2217373921, 4337012736, 4388797504, 5402250000, 5554571841, 6080256576, 7169347584, 10721359936

Offset: 1

Jonathan Sondow, Oct 28 2013

nonn

Intersection of A050802 and A038596.

Square terms of sequence A225908. - Michel Marcus, Apr 22 2016

			345744 = 588^2 = 14^3 + 70^3 = 71^3 - 23^3.

Ian Stewart, "Game, Set and Math", Dover, 2007, Chapter 8 'Close Encounters of the Fermat Kind', pp. 107-124.

Donovan Johnson and Chai Wah Wu, Table of n, a(n) for n = 1..500 n = 1..100 from Donovan Johnson
Index to sequences related to sums of cubes

Cf. A050801, A050802, A038596, A038597, A225908, A230716.

isA038596(n)=for(k=sqrtnint(n,3)+1,(sqrtint(12*n-3)+3)\6,if(ispower(n-k^3,3), return(issquare(n)))); 0
isA050802(n)=for(k=sqrtnint((n+1)\2, 3), sqrtnint(n-1, 3), if(ispower(n-k^3, 3), return(issquare(n)))); 0
is(n)=isA038596(n) && isA050802(n) \\ Charles R Greathouse IV, Oct 28 2013

a(n) = k^2 = a^3 + b^3 = c^3 - d^3 for some natural numbers k, a, b, c, d.

a(n) = A230716(n)^2.

a(5)-a(24) from Donovan Johnson, Oct 28 2013

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A023042 Numbers whose cube is the sum of three distinct nonnegative cubes.

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A226903 Shiraishi numbers: a parametrized family of solutions c to the Diophantine equation a^3 + b^3 + c^3 = d^3 with d = c+1.

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A225909 Numbers that are both a sum of two positive cubes and a difference of two consecutive cubes.

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A230717 Squares that are both a sum and a difference of two positive cubes.

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