A332883 If n = Product (p_j^k_j) then a(n) = denominator of Product (1 + 1/p_j^k_j).

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 3, 13, 7, 5, 16, 17, 3, 19, 2, 21, 11, 23, 2, 25, 13, 27, 7, 29, 5, 31, 32, 11, 17, 35, 18, 37, 19, 39, 20, 41, 7, 43, 11, 3, 23, 47, 12, 49, 25, 17, 26, 53, 9, 55, 7, 57, 29, 59, 1, 61, 31, 63, 64, 65, 11, 67, 34, 23, 35

Offset: 1

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

• nonn
• frac

Comments

Denominator of sum of reciprocals of unitary divisors of n.

Examples

			1, 3/2, 4/3, 5/4, 6/5, 2, 8/7, 9/8, 10/9, 9/5, 12/11, 5/3, 14/13, 12/7, 8/5, 17/16, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Eric Weisstein's World of Mathematics, Unitary Divisor

Crossrefs

Cf. A007947, A017666, A034448, A077610, A319677, A323166, A327158 (positions of 1's), A332881, A332882 (numerators).

Programs

Maple

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

Mathematica

Table[If[n == 1, 1, Times @@ (1 + 1/#[[1]]^#[[2]] & /@ FactorInteger[n])], {n, 1, 70}] // Denominator
Table[Sum[If[GCD[d, n/d] == 1,  1/d, 0], {d, Divisors[n]}], {n, 1, 70}] // Denominator

PARI

a(n) = denominator(sumdiv(n, d, if (gcd(d, n/d)==1, 1/d))); \\ Michel Marcus, Feb 28 2020

Formula

a(n) = denominator of Sum_{d|n, gcd(d, n/d) = 1} 1/d.

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

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

a(n) = n / A323166(n). - Antti Karttunen, Nov 13 2021