%I #14 Apr 08 2022 05:58:45
%S 7,9,12,36,700,752,4081,18745,167940,1095876,3393680,5621764,9950500,
%T 30108516
%N Numbers k such that the sum of k, the divisors of k, and the numbers < k that are coprime to k, is a perfect power.
%C Numbers k such that k + A000203(k) + A023896(k) is in A001597.
%e a(3) = 12 is a term because 12 + A000203(12) + A023896(12) = 12 + 24 + 28 = 64 = 2^6 is a perfect power.
%p f:= n -> n + n*numtheory:-phi(n)/2 + numtheory:-sigma(n):
%p g:= proc(n) local t;
%p igcd(t[2], t=ifactors(n)[2]) > 1
%p end proc:
%p select(t -> g(f(t)), [$2..2*10^6]);
%t ppQ[n_] := GCD @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[10^7], ppQ[# + DivisorSigma[1, #] + #*EulerPhi[#]/2] &] (* _Amiram Eldar_, Apr 07 2022 *)
%Y Cf. A000203, A001597, A023896.
%K nonn,more
%O 1,1
%A _J. M. Bergot_ and _Robert Israel_, Apr 06 2022