A332881 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j).

1, 2, 3, 2, 5, 1, 7, 2, 3, 5, 11, 1, 13, 7, 5, 2, 17, 1, 19, 5, 21, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 35, 1, 37, 19, 39, 5, 41, 7, 43, 11, 5, 23, 47, 1, 7, 5, 17, 13, 53, 1, 55, 7, 57, 29, 59, 5, 61, 31, 21, 2, 65, 11, 67, 17, 23, 35

Offset: 1

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

• nonn
• frac

Comments

Denominator of sum of reciprocals of squarefree divisors of n.

Examples

			1, 3/2, 4/3, 3/2, 6/5, 2, 8/7, 3/2, 4/3, 9/5, 12/11, 2, 14/13, 12/7, 8/5, 3/2, 18/17, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A001615, A008683, A017666, A048250, A007947, A109395, A187778 (positions of 1's), A306695, A308443, A308462, A332880 (numerators), A332883.

Programs

Maple

a:= n-> denom(mul(1+1/i[1], i=ifactors(n)[2])):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 28 2020

Mathematica

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Denominator

PARI

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615
A332881(n) = denominator(A001615(n)/n);

Formula

Denominators of coefficients in expansion of Sum_{k>=1} mu(k)^2*x^k/(k*(1 - x^k)).

a(n) = denominator of Sum_{d|n} mu(d)^2/d.

a(n) = denominator of psi(n)/n.

a(p) = p, where p is prime.

a(n) = n / A306695(n) = n / gcd(n, A001615(n)). - Antti Karttunen, Nov 15 2021