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

1,4

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.

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).

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A351149 A191523 A132890 * A295295 A365336 A365404

Adjacent sequences: A370780 A370781 A370782 * A370784 A370785 A370786

Keyword

nonn,easy,frac

Author

Amiram Eldar, Mar 02 2024

Status

approved