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%I #7 Sep 08 2022 08:45:28
%S 4,8,32,36,121,128,144,216,256,324,400,512,576,784,900,961,1024,1296,
%T 1600,1728,1764,1936,2304,2704,2744,2916,3136,3600,3844,4096,4356,
%U 4624,4900,5184,5776,5832,6084,6400,7056,7744,7776,8000,8100,8192,9216,9604
%N Perfect powers pp such that sigma(n)=pp for some positive integer n.
%e a(1)=4 since sigma(3)=4=2^2.
%p with(numtheory); egcd := proc(n) local L; if n>1 then L:=ifactors(n)[2]; L:=map(z->z[2],L); return igcd(op(L)) else return 1 fi; end; L:=[]: P:={}: for w to 1 do for n from 1 to 10000 do s:=sigma(n); if egcd(s)>1 then print(n,s,ifactor(s)); L:=[op(L),n]; P:=P union {s}; fi od od; L; P;
%t powerQ[n_] := Block[{pf = FactorInteger@ n, min}, min = Min @@ Last /@ pf; min > 1 && AllTrue[Last /@ pf/min, IntegerQ]]; lim = 10000; Intersection[Select[Range@ lim, powerQ], DeleteDuplicates@ Sort[DivisorSigma[1, #] & /@ Range@ lim]] (* _Michael De Vlieger_, Mar 10 2015 *)
%o (Magma) Set(Sort([SumOfDivisors(k): k in[1..10000], b in [2..15], a in [2..100] | SumOfDivisors(k) eq a^b])) // _Jaroslav Krizek_, Mar 10 2015
%o (Magma) Set(Sort([SumOfDivisors(k): k in[A065496(n)]])) // _Jaroslav Krizek_, Mar 10 2015
%Y Cf. A001597, A065496.
%K nonn
%O 1,1
%A _Walter Kehowski_, Dec 01 2006
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