A072808 Smallest m such that sigma(m) mod phi(m) = n or 0 if no solution exists.

4, 5, 8, 24, 0, 22, 16, 21, 450, 40, 25, 48, 50, 136, 32, 110, 100, 90, 144, 88, 0, 656, 121, 102, 0, 80, 169, 96, 0, 68, 64, 55, 676, 464, 289, 65, 0, 117, 162, 91, 0, 116, 225, 85, 0, 272, 529, 95, 0, 148, 288, 133, 0, 164, 0, 115, 0, 160, 841, 147, 0, 333, 128, 247

Offset: 1

Author

Labos Elemer, Jul 12 2002

Keywords

• nonn

Comments

Warning: It is only conjectured that there are no solutions for n such that a(n) = 0. The search for solutions tested all m <= 10^10 for these n.

For odd remainders a(n) is a square or twice a square. See A028982, except terms 1 and 2.

All zeros corresponding to odd terms a(n) with n < 64 confirmed up to m <= 10^24. - Giovanni Resta, Apr 02 2020

Examples

			For n=4: a(4)=24 since sigma(24)=60, phi(24)=8 and Mod(60, 8)=4.
		

Crossrefs

Cf. A000203, A000010, A028982, A063514, A067192, A067193, A071390.

Programs

Maple

V:= Vector(100):
for m from 2 to 10^7 do
 v:= numtheory:-sigma(m) mod numtheory:-phi(m);
 if v > 0 and v <= 100 and V[v] = 0 then V[v]:= m fi
od:
convert(V,list); # Robert Israel, Nov 30 2024

Mathematica

f[x_] := Mod[DivisorSigma[1, x], EulerPhi[x]]; t=Table[0, {100}]; Do[s=f[n]; If[s<101&&t[[s]]==0, t[[s]]=n], {n, 1, 10000000000}]; t

Formula

a(n) = Min{x; Mod(A000203(x), A000010(x))=n} or 0 if no solutions.

Extensions

Name corrected by Sean A. Irvine, Oct 30 2024

Name corrected by Robert Israel, Nov 30 2024