cp's OEIS Frontend

A386378 Integers x such that there exist four integers 00 and w>0 such that sigma(x)^3 = sigma(y)^3 = sigma(z)^3 = x^3 + y^3 + z^3 + t^3 + w^3.

%I A386378 #17 Jul 27 2025 18:42:47
%S A386378 30,62,90,174,238,357,390,440,495,552,762,870,894,924,1056,1146,1248,
%T A386378 1386,1560,1740,1770,1782,1824,1880,1938,1992,2046,2208,2262,2472,
%U A386378 2568,2625,2670,2686,2730,2840,2856,3000,3190,3382,3630,3666,3720,3738,3828,3885,3960,3984
%N 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.
%C A386378 The numbers x, y, z, t and w form a sigma-cubic quintuple. See Dimitrov link.
%H A386378 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A386378 S. I. Dimitrov, <a href="https://arxiv.org/abs/2408.07387">Generalizations of amicable numbers</a>, arXiv:2408.07387 [math.NT], 2024.
%e A386378 (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.
%o A386378 (PARI) is23(n) = my(z); for(k=1, sqrtnint(n\2, 3), ispower(n-k^3, 3, &z) && return([k,z]));
%o A386378 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
%Y A386378 Cf. A000203, A003325, A003328, A385325, A385397.
%K A386378 nonn
%O A386378 1,1
%A A386378 _S. I. Dimitrov_, Jul 20 2025
%E A386378 Corrected and extended by _Michel Marcus_, Jul 22 2025