A385397 Numbers x such that there exist three integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = x^3 + y^3 + z^3 + w^3.

153, 216, 255, 324, 672, 735, 1074, 1170, 1218, 2430, 2655, 2736, 3482, 4148, 4605, 4935, 5220, 5446, 5916, 6048, 7140, 9340, 11000, 11160, 12768, 14090, 14098, 14980, 17220, 17696, 18984, 21068, 21948, 22128, 23022, 23205, 24297, 24570, 25284, 25740, 29058, 29640, 30240, 30690, 31008, 31190, 32760, 37140, 39840

Offset: 1

Author

S. I. Dimitrov, Jun 27 2025

Keywords

• nonn
• hard

Comments

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

Examples

			(255, 321, 84, 312) is such a quadruple because sigma(255)^3 = sigma(321)^3 = 432^3 = 255^3 + 321^3 + 84^3 + 312^3.
		

Links

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

Crossrefs

Cf. A000203, A385325, A385356.

Programs

PARI

issc(n) = if (n>0, for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3) && return(1))); \\ A003325
isok(x) = my(s=sigma(x)); for (y=1, x, if (s == sigma(y), if (issc(s^3-x^3-y^3), return(1)););); \\ Michel Marcus, Jun 27 2025

Extensions

More terms from David A. Corneth, Jun 27 2025