A076495 Smallest x such that sigma(x) mod x = n, or 0 if no such x exists.

2, 20, 4, 9, 0, 25, 8, 10, 15, 14, 21, 24, 27, 22, 16, 26, 39, 208, 36, 34, 51, 38, 57, 112, 95, 46, 69, 48, 115, 841, 32, 58, 45, 62, 93, 660, 155, 1369, 162, 44, 63, 1681, 50, 82, 123, 52, 129, 60, 75, 94, 72, 352, 235, 90, 329, 84, 99, 68, 265, 96, 371, 118, 64, 76

Offset: 1

Author

Labos Elemer, Oct 21 2002

Keywords

• nonn

Comments

At present, the 0 entry for n=5 is only a conjecture.

For n <= 1000, a(5) and a(898) are the only terms not found using x <= 10^11. - Donovan Johnson, Sep 20 2012

10^11 < a(898) <= 140729946996736. - Donovan Johnson, Sep 28 2013

a(898) > 10^13 and the same bound holds for a(5), if it exists. - Giovanni Resta, Apr 02 2014

a(5) > 1.5*10^14, if it exists. - Jud McCranie, Jun 02 2019

Examples

			n=1: a(1) = smallest prime = 2.
n=3: a(3) = 4 since sigma(4) mod 4 = 7 mod 4 = 3.
n=5: Very difficult case (see Comments section).
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..1000
• Carl Pomerance, On the congruences σ(n) ≡ a (mod n) and n ≡ a (mod φ(n)), Acta Arithmetica 26:3 (1974-1975), pp. 265-272. (See theorem 4.)

Crossrefs

Cf. A045768, A045769, A045770, A054024.

Programs

Mathematica

f[x_] := s=Mod[DivisorSigma[1, n], n]; t=Table[0, {256}]; Do[s=f[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000}]; t

PARI

a(n)=my(k);while(sigma(k++)%k!=n,);k \\ Charles R Greathouse IV, Dec 28 2013