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A376119 Abundant numbers that are perfect powers.

%I A376119 #15 Dec 12 2024 18:24:40
%S A376119 36,100,144,196,216,324,400,576,784,900,1000,1296,1600,1728,1764,1936,
%T A376119 2304,2500,2704,2744,2916,3136,3600,4356,4624,4900,5184,5776,5832,
%U A376119 6084,6400,7056,7744,7776,8000,8100,8464,9216,9604,10000,10404,10648,10816,11025,11664,12100,12544
%N A376119 Abundant numbers that are perfect powers.
%C A376119 Intersection of A001597 and A005101.
%H A376119 Robert Israel, <a href="/A376119/b376119.txt">Table of n, a(n) for n = 1..10000</a>
%e A376119 36 is a term being a power (36=6^2) and an abundant number as a multiple of 6.
%p A376119 N:= 20000: # to get terms <= N
%p A376119 isab:= proc(n) local F, t;
%p A376119   F:= ifactors(n)[2];
%p A376119   mul((t[1]^(t[2]+1)-1)/(t[1]-1), t = F) > 2*n
%p A376119 end proc:
%p A376119 S:= select(isab, {seq(seq(x^i,i=2..ilog[x](N)),x=2..isqrt(N))}):
%p A376119 sort(convert(S,list)); # _Robert Israel_, Sep 12 2024
%t A376119 Select[Range[2*10^4], DivisorSigma[1,#]-2#>0&&GCD@@FactorInteger[#][[All, 2]]>1&]
%o A376119 (PARI) ok(n)=sigma(n)-2*n>0 && ispower(n)
%Y A376119 Cf. A001597, A005101.
%K A376119 nonn
%O A376119 1,1
%A A376119 _Waldemar Puszkarz_, Sep 11 2024