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%I #17 Mar 08 2023 14:58:55
%S 119,735,5145,36015
%N Numbers k such that the ratio of A117731(k) and A082687(k) is composite.
%C a(1) = 7*17, a(2) = 3*5*7^2, a(3) = 3*5*7^3.
%C Corresponding composite terms in A125741 are {119, 49, 49, ...}.
%C A125741(n) is composite for n = {7, 16, 36, ...}.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HilbertMatrix.html">Hilbert Matrix</a>.
%t h=0; Do[ h=h+1/(n+1)/(2n+1); f=Numerator[n*h]; g=Numerator[h]; If[ !Equal[f, g] && !PrimeQ[f/g], Print[ {n, f/g, FactorInteger[n], FactorInteger[f/g]} ] ], {n, 1, 10000} ]
%o (PARI) f(n) = sum(k=1, n, 1/(n+k));
%o isok(k) = my(fk = f(k), q = numerator(k*fk)/ numerator(fk)); (q!=1) && !isprime(q); \\ _Michel Marcus_, Mar 08 2023
%Y Cf. A117731, A082687, A125740, A125741, A126196, A126197, A125581.
%K bref,hard,more,nonn
%O 1,1
%A _Alexander Adamchuk_, Mar 12 2007, Jun 09 2007
%E Edited by _Max Alekseyev_, Jul 12 2019
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