A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Mar 02 2024

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with: 1, 1, 1, 3/2, 1, 1, 1, 3/2, 4/3, 1, 1, 3/2, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Crossrefs

Cf. A005117, A057521, A157289, A295295, A332880, A370784 (denominators).

Programs

Mathematica

a[n_] := Numerator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + 1/f[i,1])));}

Formula

a(n) = A332880(A057521(n)).

Let f(n) = a(n)/A370784(n):

f(n) is multiplicative with f(p) = 1 and f(p^e) = 1 + 1/p for e >= 2.

f(n) = 1 if and only if n is squarefree (A005117).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = zeta(3)/zeta(6) = 1.181564... (A157289) (Jakimczuk, 2024).