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A213931 Smallest number k such that the sum of divisors of k equals n times a nontrivial integer power.

%I A213931 #18 Nov 21 2020 16:16:33
%S A213931 3,7,6,21,19,14,12,21,22,27,43,33,63,28,24,66,67,30,98,57,44,129,367,
%T A213931 42,199,63,85,84,463,54,48,93,86,201,76,66,219,111,99,120,163,60,1285,
%U A213931 129,88,274,751,105,156,199,134,198,211,102,327,84,147,346,1765
%N A213931 Smallest number k such that the sum of divisors of k equals n times a nontrivial integer power.
%C A213931 Smallest k such that sigma(k) = n * m^q where m, q >= 2.
%H A213931 Alois P. Heinz, <a href="/A213931/b213931.txt">Table of n, a(n) for n = 1..5000</a>
%e A213931 a(34) = 201 because sigma(201) = 272 = 34*2^3.
%p A213931 with(numtheory):
%p A213931 a:= proc(n) local k, q;
%p A213931       for k while irem(sigma(k), n, 'q')>0 or
%p A213931       igcd(map(i->i[2], ifactors(q)[2])[])<2 do od; k
%p A213931     end:
%p A213931 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 26 2012
%t A213931 a[n_] := Module[{k, q, r}, For[k = 1, {q, r} = QuotientRemainder[ DivisorSigma[1, k], n]; r>0 || GCD @@ FactorInteger[q][[All, 2]]<2, k++]; k];
%t A213931 Array[a, 100] (* _Jean-François Alcover_, Nov 21 2020, after _Alois P. Heinz_ *)
%o A213931 (PARI) a(n)=my(k);while(sigma(k++)%n || !ispower(sigma(k)/n), ); k \\ _Charles R Greathouse IV_, Jun 26 2012
%Y A213931 Cf. A000203, A001597, A213401, A213386, A213420.
%K A213931 nonn
%O A213931 1,1
%A A213931 _Michel Lagneau_, Jun 25 2012