A385531 Numbers x such that there exist three integers 0<x<=y<=z and t>0 such that sigma(x)^2 = sigma(y)^2 = sigma(z)^2 = x^2 + y^2 + z^2 + t^2.

4, 6, 28, 45, 48, 60, 156, 204, 208, 360, 496, 1170, 2016, 2520, 2925, 3480, 4796, 5532, 5733, 7152, 7605, 8128, 9680, 11050, 12402, 15776, 33468, 36720, 37064, 38408, 43584, 50960, 55216, 63708, 70364, 83772, 92280, 106700, 114840, 116288, 149400, 163800, 166617, 167580

Offset

1,1

Comments

The numbers x, y, z and t form a sigma-quadratic quadruple. See Dimitrov link.

Links

David A. Corneth, Table of n, a(n) for n = 1..97

Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

David A. Corneth, PARI program

S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Example

(3480, 3672, 4296, 8520) is such a quadruple because sigma(3480)^2 = sigma(3672)^2 = sigma(4296)^2 = 3480^2 + 3672^2 + 4296^2 + 8520^2.

Prog

(PARI) isok(x) = my(s=sigma(x), vi=select(t->(t>=x), invsigma(s))); for (i=1, #vi, for (j=1, #vi, for (k=1, #vi, if ((i==1) || (j==1) || (k==1), my(ss = s^2 - vi[i]^2 - vi[j]^2 - vi[k]^2); if (ss && issquare(ss), return(1)); ); ); ); ); \\ Michel Marcus, Jul 09 2025
(PARI) \\ See Corneth link

Crossrefs

Cf. A000203, A000414, A385356, A385397.

Sequence in context: A106286 A066293 A204385 * A385810 A341045 A050881

Adjacent sequences: A385528 A385529 A385530 * A385532 A385533 A385534

Keyword

nonn,hard

Author

S. I. Dimitrov, Jul 02 2025

Extensions

Some missing terms added by Michel Marcus, Jul 09 2025

More terms from David A. Corneth, Jul 09 2025

Status

approved