A332880 If n = Product (p_j^k_j) then a(n) = numerator of Product (1 + 1/p_j).

1, 3, 4, 3, 6, 2, 8, 3, 4, 9, 12, 2, 14, 12, 8, 3, 18, 2, 20, 9, 32, 18, 24, 2, 6, 21, 4, 12, 30, 12, 32, 3, 16, 27, 48, 2, 38, 30, 56, 9, 42, 16, 44, 18, 8, 36, 48, 2, 8, 9, 24, 21, 54, 2, 72, 12, 80, 45, 60, 12, 62, 48, 32, 3, 84, 24, 68, 27, 32, 72

Offset: 1

Author

Ilya Gutkovskiy, Feb 28 2020

Keywords

• nonn
• frac

Comments

Numerator of sum of reciprocals of squarefree divisors of n.

(6/Pi^2) * A332881(n)/a(n) is the asymptotic density of numbers that are coprime to their digital sum in base n+1 (see A094387 and A339076 for bases 2 and 10). - Amiram Eldar, Nov 24 2022

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, A017665, A028235, A028236, A048250, A076512, A306695, A308443, A308462, A332881 (denominators), A332882.

Cf. also A348734, A348735.

Cf. A059956, A065463, A082020, A094387, A339076.

Programs

Maple

a:= n-> numer(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}] // Numerator
Table[Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}], {n, 1, 70}] // Numerator

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
A332880(n) = numerator(A001615(n)/n);

Formula

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

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

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

a(p) = p + 1, where p is prime.

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

From Amiram Eldar, Nov 24 2022: (Start)

Asymptotic means:

Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A332881(k) = 15/Pi^2 = 1.519817... (A082020).

Limit_{m->oo} (1/m) * Sum_{k=1..m} A332881(k)/a(k) = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442... (A065463). (End)