A386225 Numbers x such that there exist four integers 00, t>0 and w>0 such that sigma(x)^4 = sigma(y)^4 = x^4 + y^4 + z^4 + t^4 + w^4.

24, 240, 600

Offset: 1

Author

S. I. Dimitrov, Jul 15 2025

Keywords

• nonn
• hard
• bref
• more

Comments

The numbers x, y, z, t and w form a sigma-quartic quintuple.

[91963648, 91963648, 137945472, 183927296, 183927296] is another quintuple. - Michel Marcus, Jul 28 2025

Examples

			(24, 24, 36, 48, 48) is such a quintuple because sigma(24)^4 = sigma(24)^4 = 60^4 = 24^4 + 24^4 + 36^4 + 48^4 + 48^4.
(240, 240, 240, 408, 720) and (600, 600, 600, 1020, 1800) are the two next quintuples.
		

Links

• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
• S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Crossrefs

Cf. A000203, A003339, A297305, A385397.

Programs

PARI

find4(ss) = my(v=List(), k, t); ss\=1; for(x=1, sqrtnint(ss-2, 4), for(y=1, min(sqrtnint(ss-x^4-1, 4), x), k=x^4+y^4; for(z=1, min(sqrtnint(ss-k, 4), y), if (k+z^4==ss, return([x,y,z])))));
isok4(x) = my(s=sigma(x), v=select(z->(z>=x), invsigma(s))); if (#v >=2, for (i=1, #v, my(k=s^4 - x^4 - v[i]^4); if (k>0, my(xyz = find4(k)); if (xyz, return([x, v[i], xyz[1], xyz[2], xyz[3]]));););); \\ Michel Marcus, Jul 22 2025