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A272981 Least prime k>1 such that the sum of divisors of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

%I A272981 #13 May 17 2016 09:12:36
%S A272981 3,7,7,31,31,211,211,211,211,2311,2311,120121,120121,120121,120121,
%T A272981 4084081,4084081,106696591,106696591,106696591,106696591,892371481,
%U A272981 892371481,892371481,892371481,892371481,892371481,71166625531,71166625531,200560490131,200560490131
%N A272981 Least prime k>1 such that the sum of divisors of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).
%C A272981 For 1<n<11 A272981(n) = A092967(n+1).
%C A272981 The different numbers are listed in A073917.
%H A272981 Paolo P. Lava, <a href="/A272981/b272981.txt">Table of n, a(n) for n = 1..100</a>
%e A272981 sigma(3) / d(3) = 4 / 2 = 2 but sigma(3^2) / d(3^2) = 13 / 3;
%e A272981 sigma(7) / d(7) = 8 / 2 = 4, sigma(7^2) / d(7^2) = 57 / 3 = 19, sigma(7^3) / d(7^3) = 400 / 4 = 100 but sigma(7^4) / d(7^4) = 2801 / 5.
%p A272981 with(numtheory): P:= proc(q) local a,j,k,ok,p; global n; a:=2;
%p A272981 for k from 1 to q do for n from a to q do ok:=1;
%p A272981 for j from 1 to k do if not type(sigma(n^j)/tau(n^j),integer) then ok:=0; break; fi; od;
%p A272981 if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);
%t A272981 Table[SelectFirst[Range[2, 10^6], AllTrue[#^Range@ n, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] &], {n, 15}] (* _Michael De Vlieger_, May 12 2016, Version 10 *)
%Y A272981 Cf. A000005, A000203, A073917, A092967, A272980.
%K A272981 nonn
%O A272981 1,1
%A A272981 _Paolo P. Lava_, May 12 2016