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

Author

M. F. Hasler, May 14 2015

Keywords

• nonn

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.

Programs

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 *)

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.

Extensions

Corrected and extended by Giovanni Resta, May 18 2015