A396309 Integers x such that there exist two integers 0<y<=z such that sigma(x)^2*psi(x) = x^3 + y^3 + z^3.

5, 6, 53, 58, 102, 112, 118, 196, 197, 214, 224, 248, 258, 426, 445, 487, 491, 861, 885, 1182, 1299, 1615, 2082, 2190, 2465, 2519, 2679, 3994, 4035, 4048, 4210, 4583, 5326, 5506, 5673, 5690, 6101, 6237, 6454, 7778, 7953, 8062, 8429, 8585, 9095, 10194, 11399, 13015, 14464, 14723, 16937, 17790, 18797

Offset

1,1

Comments

The numbers x, y and z form a sigma^2*psi-cubic triple.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..447

S. I. Dimitrov, On σψ-quadratic k-tuples and related generalizations, hal-05303937, 2025.

Example

(196, 245, 315) is such a triple because sigma(196)^2 * psi(196) = 399^2*336 = 196^3 + 245^3 + 315^3.

Prog

(PARI) dedekindpsi(n)=n*prod(p=factor(n)[, 1], 1+1/p)
isok(x)=
{
  my(d = sigma(x)^2 * dedekindpsi(x) - x^3);
  for(y = 1, sqrtnint(d, 3),
    if((t = d - y^3) < y^3, break);
    if(ispower(t, 3), return(1))
  );
  0
}
(PARI) T=thueinit('z^3+1);
isSumOfTwoCubes(n)=#select(v->min(v[1], v[2])>0, thue(T, n))>0
is(n, f=factor(n))=isSumOfTwoCubes(n*sigma(f)^2*sumdivmult(f, d, issquarefree(d)/d) - n^3) \\ Charles R Greathouse IV, May 22 2026

Crossrefs

Cf. A000203, A001615, A003072, A385325, A391290, A003325.

Sequence in context: A163481 A298376 A269908 * A385325 A157805 A256291

Adjacent sequences: A396306 A396307 A396308 * A396310 A396311 A396312

Keyword

nonn

Author

S. I. Dimitrov, May 21 2026

Status

approved