A386378 Integers x such that there exist four integers 0<x<=y<=z, t>0 and w>0 such that sigma(x)^3 = sigma(y)^3 = sigma(z)^3 = x^3 + y^3 + z^3 + t^3 + w^3.

30, 62, 90, 174, 238, 357, 390, 440, 495, 552, 762, 870, 894, 924, 1056, 1146, 1248, 1386, 1560, 1740, 1770, 1782, 1824, 1880, 1938, 1992, 2046, 2208, 2262, 2472, 2568, 2625, 2670, 2686, 2730, 2840, 2856, 3000, 3190, 3382, 3630, 3666, 3720, 3738, 3828, 3885, 3960, 3984

Offset

1,1

Comments

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

Links

Table of n, a(n) for n=1..48.

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

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

Example

(174, 190, 323, 5, 94) is such a quintuple because sigma(174)^3 = sigma(190)^3 = sigma(323)^3 = 360^3 = 174^3 + 190^3 + 323^3 + 5^3 + 94^3.

Prog

(PARI) is23(n) = my(z); for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3, &z) && return([k, z]));
isok3(x) = my(s=sigma(x), v=select(z->(z>=x), invsigma(s))); if (#v >= 1, for (i=1, #v, for (j=1, #v, my(k=s^3 - x^3 - v[i]^3-v[j]^3); if (k>0, my(tw = is23(k)); if (tw, return([x, v[i], v[j], tw[1], tw[2]])); ); ); ); ); \\ Michel Marcus, Jul 22 2025

Crossrefs

Cf. A000203, A003325, A003328, A385325, A385397.

Sequence in context: A042796 A042798 A320705 * A132771 A396564 A326158

Adjacent sequences: A386375 A386376 A386377 * A386379 A386380 A386381

Keyword

nonn

Author

S. I. Dimitrov, Jul 20 2025

Extensions

Corrected and extended by Michel Marcus, Jul 22 2025

Status

approved