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A078538 Smallest k > 6 such that sigma_n(k)/phi(k) is an integer.

%I A078538 #24 Jun 27 2025 21:41:14
%S A078538 12,22,12,249,12,22,12,19689,12,22,12,249,12,22,12
%N A078538 Smallest k > 6 such that sigma_n(k)/phi(k) is an integer.
%C A078538 For n = 16, 48, 64, and 80 the solutions are hard to find, exceed 10^6 or even 10^7.
%C A078538 If a(16) exists, it is greater than 2^32. Terms a(17) to a(47) are 12, 22, 12, 249, 12, 22, 12, 9897, 12, 22, 12, 249, 12, 22, 12, 2566, 12, 22, 12, 249, 12, 22, 12, 19689, 12, 22, 12, 249, 12, 22, 12. - _T. D. Noe_, Dec 08 2013
%e A078538 These terms appear as 5th entries in A020492, A015759-A015774. k = {1, 2, 3, 6} are solutions to Min{k : Mod[sigma[n, k], phi[k]]=0}. First nontrivial solutions are larger: for odd n, k = 12 is solution; for even powers larger numbers arise like 22, 249, 9897, 19689, etc. Certain power-sums of divisors proved to be hard to find.
%t A078538 f[k_, x_] := DivisorSigma[k, x]/EulerPhi[x]; Table[k=7; While[!IntegerQ[f[n, k]], k++]; k, {n, 1, 15}] (* corrected by _Jason Yuen_, Jun 27 2025 *)
%o A078538 (PARI) ok(n,k)=my(f=factor(n), r=sigma(f,k)/eulerphi(f)); r>=7 && denominator(r)==1
%o A078538 a(n)=my(k=7); while(!ok(k, n), k++); k \\ _Charles R Greathouse IV_, Nov 27 2013
%o A078538 (Python)
%o A078538 from sympy import divisors, totient as phi
%o A078538 def a(n):
%o A078538     k, pk = 7, phi(7)
%o A078538     while sum(pow(d, n, pk) for d in divisors(k, generator=True))%pk != 0:
%o A078538         k += 1
%o A078538         pk = phi(k)
%o A078538     return k
%o A078538 print([a(n) for n in range(1, 16)]) # _Michael S. Branicky_, Dec 22 2021
%Y A078538 Cf. A000203, A001157, A001158, A000010, A015759-A015774, A020492.
%K A078538 hard,more,nonn
%O A078538 1,1
%A A078538 _Labos Elemer_, Nov 29 2002