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).

1, 0, 3, 7, 21, 63, 252, 945, 4095, 0, 110565, 515970, 3611790, 21670740, 130024440, 806151528, 7255363752, 47159864388, 471598643880, 3301190507160, 26409524057280, 237685716515520, 2852228598186240, 21391714486396800

Offset: 1

Author

Leroy Quet, Aug 22 2005

Keywords

• nonn

Comments

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?

No other nonintegers found up to 200000. - Michel Marcus, Sep 14 2015

No other nonintegers up to 3000000. - Robert Israel, Jan 22 2018

Examples

			a(4) = 1 * 3 * 4 * 7 /(1 * 2 * 2 * 3) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..558

Crossrefs

Cf. A000005, A000203.

Cf. A066780, A066843.

Programs

Maple

p:= 1: A[1]:= 1:
for n from 2 to 50 do
  p:= p * numtheory:-sigma(n)/numtheory:-tau(n);
  if p::integer then A[n]:= p else A[n]:= 0 fi
od:
seq(A[n],n=1..50); # Robert Israel, Jan 22 2018

Mathematica

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 *)

PARI

a(n) = my(q = prod(k=1, n, sigma(k)/numdiv(k))); if (denominator(q)==1, q, 0); \\ Michel Marcus, Sep 14 2015

Formula

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).

a(n) = A066780(n)/A066843(n) if this is an integer, else 0. - Michel Marcus, Sep 14 2015

Extensions

More terms from Stefan Steinerberger, Oct 24 2007