A390296 Numbers k such that sigma(k) = psi(k) + tau(k)^2.

460, 476, 846, 1035, 1071, 1380, 1428, 1540, 2350, 2790, 2898, 2975, 3465, 4606, 4620, 5635, 7050, 8050, 8925, 11374, 13818, 13915, 14399, 15190, 15886, 16905, 19435, 20111, 24150, 27166, 33235, 33934, 34122, 37510, 38962, 41515, 41745, 42959, 43197, 45570, 47658, 49726, 52390, 54418, 58305, 60333

Offset

1,1

Comments

Includes p * q * r^2 where p, q, r are distinct primes with (p + 1)*(q + 1) = 144. The possible {p,q} pairs are {2, 47}, {5, 23}, {7, 17}. - Robert Israel, Nov 11 2025

Links

Robert Israel, Table of n, a(n) for n = 1..1000

Example

460 is in the sequence since sigma(460) = 1008 = 864 + 12^2 = psi(460) + tau(460)^2.

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^2
end proc:
select(filter, [$1..10^5]); # Robert Israel, Nov 11 2025

Mathematica

psi[n_] := n * Times @@ (1 + 1/FactorInteger[n][[;; , 1]]); psi[1] = 1; Select[Range[61000], DivisorSigma[1, #] == psi[#] + DivisorSigma[0, #]^2 &] (* Amiram Eldar, Oct 31 2025 *)

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A001615 (psi), A387953, A387999, A389478, A390251, A390297.

Sequence in context: A048923 A329292 A252535 * A377747 A264242 A358894

Adjacent sequences: A390293 A390294 A390295 * A390297 A390298 A390299

Keyword

nonn

Author

S. I. Dimitrov, Oct 31 2025

Status

approved