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%I #14 Dec 25 2016 02:15:00
%S 4,9,20,25,169,289,961,1849,3721,6889,11881,14641,15625,17161,52441,
%T 57121,66049,69169,72361,96721,97969,117649,130321,196249,214369,
%U 253009,326041,351649,358801,383161,410881,418609,426409,434281,491401
%N Numbers n such that sigma_4(n)/sigma_2(n) is prime.
%C Except for 3rd term 20, below 10000000 all other entries are even powers of a prime. These primes are listed in A066111. It is not known if other numbers similar to 20 exist or not.
%H Harry J. Smith, <a href="/A066109/b066109.txt">Table of n, a(n) for n=1..250</a>
%F n such that A001159(n)/A001157(n) is prime.
%e m = 20: divisors[20] = {20, 10, 5, 4, 2, 1}, sigma_4 = 160000 + 10000 + 625 + 256 + 16 + 1 = 170898, sigma_2 = 400 + 100 + 25 + 16 + 4 + 1 = 546; p = 170898/546 = 73 is prime.
%t Do[s = DivisorSigma[4, n]; z = DivisorSigma[2, n]; If[PrimeQ[s/z], Print[{n, s, z, s/z}]], {n, 1, 10000000}]
%t Select[Range[500000],PrimeQ[DivisorSigma[4,#]/DivisorSigma[2,#]]&] (* _Harvey P. Dale_, May 02 2011 *)
%o (PARI) { n=0; for (m=1, 10^9, if (frac(f=sigma(m, 4)/sigma(m, 2)), next); if (isprime(f), write("b066109.txt", n++, " ", m); if (n==250, return)) ) } \\ _Harry J. Smith_, Nov 16 2009
%Y Cf. A000040, A001157, A001159.
%K nonn
%O 1,1
%A _Labos Elemer_, Dec 05 2001