A389546 Numbers k such that sigma(k) = psi(k) + tau(k)^4.

72548, 78268, 79396, 80228, 81028, 81196, 81748, 124398, 163233, 176103, 178641, 180513, 182313, 182691, 183933, 345550, 453425, 489175, 496225, 501425, 506425, 507475, 510925, 677278, 912252, 958783, 963852, 967540, 972601, 980652, 982793, 986028, 987852, 992593, 994651, 1001413, 1081940, 1094740

Offset

1,1

Comments

Includes p*q*r^2 where p, q and r are distinct primes with (p+1) * (q+1) = 20736. The allowed {p,q} pairs are {2, 6911}, {7, 2591}, {17, 1151}, {23, 863}, {31, 647}, {47, 431}, {53, 383}, and {107, 191}. - Robert Israel, Nov 11 2025

Links

Table of n, a(n) for n=1..38.

Example

72548 is in the sequence since sigma(72548) = 145152 = 124416 + 12^4 = psi(72548) + tau(72548)^4.

Maple

filter:= proc(n) local F, sigma, psi, tau, t;
 F:= ifactors(n)[2];
 sigma:= mul((t[1]^(1+t[2])-1)/(t[1]-1), t=F);
 psi:= n * mul(1+1/t[1], t=F);
 tau:= mul(1+t[2], t=F);
 sigma = psi + tau^4
end proc:
select(filter, [$1 .. 2*10^6]); # Robert Israel, Nov 11 2025

Mathematica

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Select[Range[200000], DivisorSigma[1, #] == psi[#] + DivisorSigma[0, #]^4 &] (* Amiram Eldar, Nov 05 2025 *)

Prog

(PARI) isok(k) = {my(f = factor(k)); sigma(f) == prod(i=1, #f~, (f[i, 1]+1) * f[i, 1]^(f[i, 2]-1)) + numdiv(f)^4; } \\ Amiram Eldar, Nov 05 2025

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A001615 (psi), A387953, A387999, A389478, A390252, A390296, A390297, A390420.

Sequence in context: A183788 A369238 A093212 * A114258 A391002 A238151

Adjacent sequences: A389543 A389544 A389545 * A389547 A389548 A389549

Keyword

nonn

Author

S. I. Dimitrov, Nov 05 2025

Status

approved