A072808 - OEIS

%I #48 Dec 01 2024 10:06:06

%S 4,5,8,24,0,22,16,21,450,40,25,48,50,136,32,110,100,90,144,88,0,656,

%T 121,102,0,80,169,96,0,68,64,55,676,464,289,65,0,117,162,91,0,116,225,

%U 85,0,272,529,95,0,148,288,133,0,164,0,115,0,160,841,147,0,333,128,247

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

%C 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.

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

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

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

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

%p V:= Vector(100):

%p for m from 2 to 10^7 do

%p  v:= numtheory:-sigma(m) mod numtheory:-phi(m);

%p  if v > 0 and v <= 100 and V[v] = 0 then V[v]:= m fi

%p od:

%p convert(V,list); # _Robert Israel_, Nov 30 2024

%t 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

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

%K nonn

%O 1,1

%A _Labos Elemer_, Jul 12 2002

%E Name corrected by _Sean A. Irvine_, Oct 30 2024

%E Name corrected by _Robert Israel_, Nov 30 2024