A084306 Numbers m not divisible by 6 such that sigma(m) == 12 (mod m).

121, 304, 127744, 33501184, 8589082624

Offset

1,1

Comments

If n = P*q, where P is a multiple perfect number and q is prime so that gcd(P,q) = 1, then sigma(n) = kP(q+1) = kn + kP. Consequently sigma(n) == kP (mod n). Such values of n are regular solutions to this congruence. Here P = 6, and we disallow regular solutions and some additional solutions divisible by 6. Cf. A076496 and A234238.

a(6) > 10^11. - Donovan Johnson, Sep 20 2012

If p = 2^k - 13 > 3 is a prime number, then 2^(k-1)*p is a term. This happens for k = 5, 9, 13, 17, 57, 105, 137, 3217, ... (A096818). - Giovanni Resta, Apr 01 2014

a(6) > 10^24. - Max Alekseyev, Mar 16 2026

Links

Table of n, a(n) for n=1..5.

Example

n = 33501184 = 4096*8179; sigma(n) = 2n + 12 = 67002380.

Mathematica

Do[s=Mod[DivisorSigma[1, n], n]; If[IntegerQ[n/100000], Print[{n}]]; If[Equal[s, 12]&&!Equal[Mod[n, 6], 0], Print[n]], {n, 1, 100000000}]

Crossrefs

Subsequence of A076496 and A234238.

Cf. A076495, A000396, A045768, A045769, A045770, A007691, A096818.

Sequence in context: A038467 A106562 A106573 * A254978 A254971 A112075

Adjacent sequences: A084303 A084304 A084305 * A084307 A084308 A084309

Keyword

nonn,more

Author

Labos Elemer, Jun 11 2003

Extensions

a(5) from Donovan Johnson, Sep 20 2012

Edited by Max Alekseyev, Jan 16 2026

Status

approved