A226902 -id:A226902 - cp's OEIS frontend

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.

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.

A226904 Solutions c to the Diophantine equation a^3 + b^3 + c^3 = (c+1)^3 that are not Shiraishi numbers.

Original entry on oeis.org

8, 40, 70, 114, 188, 213, 248, 255, 297, 453, 460, 477, 487, 495, 564, 632, 671, 768, 909, 958, 1190, 1324, 1331, 1346, 1744, 1779, 2068, 2130, 2208, 2262, 2448, 2790, 3320, 3327, 3414, 3466, 3764, 3866, 3973, 4154, 4193, 4378, 4473, 4583, 4645, 5033, 5175

Offset: 1

Jonathan Sondow, Jun 22 2013

nonn

Complement of A226903 in A226902.

Cf. A226902, A226903.

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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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A226904 Solutions c to the Diophantine equation a^3 + b^3 + c^3 = (c+1)^3 that are not Shiraishi numbers.

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