A257965 Numbers n such that n^3 = a^2 + b^2 and a^3 + b^3 is a square, for some positive integers a and b.

2, 32, 125, 162, 512, 1250, 2000, 2592, 4802, 8192, 10125, 13122, 20000, 29282, 32000, 41472, 49000, 57122, 76832, 78125, 92450, 101250, 131072, 152881, 162000, 167042, 207025, 209952, 215306, 260642, 300125, 320000, 388962, 468512, 512000, 559682, 663552

Offset

1,1

Comments

A subsequence of A000404.

Two subsequences are given by: n = 2*m^4 with a = b = 2*m^6, and n = 5^3*m^4 with a = 5^4*m^6, b = 2*a.

Links

Giovanni Resta, Table of n, a(n) for n = 1..526 (terms < 10^10)

Giovanni Resta, Decompositions for the first 526 terms.

Mathematica

L = {}; n = 0; Clear[x, y]; While[n < 5000, n++; If[ And @@ (EvenQ /@ Last /@ Select[FactorInteger[n], Mod[First[#] + 1, 4] == 0 &]) && False =!= (sol = Reduce[n^3 == x^2 + y^2 && x >= y > 0, {x, y}, Integers]), ss = {x, y} /. List@ToRules@sol; If[{} != Select[ss, IntegerQ@ Sqrt@ Total[#^3] &, 1], AppendTo[L, n]; Print@L]]]; L (* Giovanni Resta, May 18 2015 *)

Prog

(PARI) is(n)={is_A000404(n)&&for(i=1, #T=sum2sqr(n^3), T[i][1]&&issquare(T[i][1]^3+T[i][2]^3)&&return(T[i]))} \\ Uses sum2sqr(), cf. A133388. Returns [a, b] if n is in the sequence, 0 else.

Crossrefs

Sequence in context: A122127 A303078 A226395 * A191997 A274654 A112850

Adjacent sequences: A257962 A257963 A257964 * A257966 A257967 A257968

Keyword

nonn

Author

M. F. Hasler, May 14 2015

Extensions

Corrected and extended by Giovanni Resta, May 18 2015

Status

approved