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A109361 a(n) = Product_{k=1..n} sigma(k)/d(k), where sigma(k) = Sum_{j|k} j and d(k) = Sum_{j|k} 1. Set a(n) = 0 if the corresponding product is not an integer (e.g., for n=2 and n=10).

%I A109361 #28 Jan 23 2018 06:58:17
%S A109361 1,0,3,7,21,63,252,945,4095,0,110565,515970,3611790,21670740,
%T A109361 130024440,806151528,7255363752,47159864388,471598643880,
%U A109361 3301190507160,26409524057280,237685716515520,2852228598186240,21391714486396800
%N A109361 a(n) = Product_{k=1..n} sigma(k)/d(k), where sigma(k) = Sum_{j|k} j and d(k) = Sum_{j|k} 1. Set a(n) = 0 if the corresponding product is not an integer (e.g., for n=2 and n=10).
%C A109361 The product at n = 2 is the noninteger 1.5. The product at n = 10 is the noninteger 18427.5. Jack Brennen's observed that the only values which are not integers occur when n = 2 or 10, for n < 5000. Are all products for n >= 11 integers?
%C A109361 No other nonintegers found up to 200000. - _Michel Marcus_, Sep 14 2015
%C A109361 No other nonintegers up to 3000000. - _Robert Israel_, Jan 22 2018
%H A109361 Robert Israel, <a href="/A109361/b109361.txt">Table of n, a(n) for n = 1..558</a>
%F A109361 Product_{k=1..n} sigma(k)/d(k) = Product_{p=primes} Product_{k>=1} ((p^(k+1)-1)*k/((p^k -1)(k+1)))^floor(n/p^k).
%F A109361 a(n) = A066780(n)/A066843(n) if this is an integer, else 0. - _Michel Marcus_, Sep 14 2015
%e A109361 a(4) = 1 * 3 * 4 * 7 /(1 * 2 * 2 * 3) = 7.
%p A109361 p:= 1: A[1]:= 1:
%p A109361 for n from 2 to 50 do
%p A109361   p:= p * numtheory:-sigma(n)/numtheory:-tau(n);
%p A109361   if p::integer then A[n]:= p else A[n]:= 0 fi
%p A109361 od:
%p A109361 seq(A[n],n=1..50); # _Robert Israel_, Jan 22 2018
%t A109361 Table[If[IntegerQ[Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}]], Product[DivisorSigma[1, k]/Length[Divisors[k]], {k, 1, n}], 0], {n, 1, 30}] (* _Stefan Steinerberger_, Oct 24 2007 *)
%o A109361 (PARI) a(n) = my(q = prod(k=1, n, sigma(k)/numdiv(k))); if (denominator(q)==1, q, 0); \\ _Michel Marcus_, Sep 14 2015
%Y A109361 Cf. A000005, A000203.
%Y A109361 Cf. A066780, A066843.
%K A109361 nonn
%O A109361 1,3
%A A109361 _Leroy Quet_, Aug 22 2005
%E A109361 More terms from _Stefan Steinerberger_, Oct 24 2007