A343443 If n = Product (p_j^k_j) then a(n) = Product (k_j + 2), with a(1) = 1.

1, 3, 3, 4, 3, 9, 3, 5, 4, 9, 3, 12, 3, 9, 9, 6, 3, 12, 3, 12, 9, 9, 3, 15, 4, 9, 5, 12, 3, 27, 3, 7, 9, 9, 9, 16, 3, 9, 9, 15, 3, 27, 3, 12, 12, 9, 3, 18, 4, 12, 9, 12, 3, 15, 9, 15, 9, 9, 3, 36, 3, 9, 12, 8, 9, 27, 3, 12, 9, 27, 3, 20, 3, 9, 12, 12, 9, 27, 3, 18

Offset: 1

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

• nonn
• easy
• mult

Comments

Inverse Moebius transform of A056671.

a(n) depends only on the prime signature of n (see formulas). - Bernard Schott, May 03 2021

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A001221, A005361, A007425, A025847, A034444, A056671, A064549, A074816, A107758, A107759, A348018, A363194.

Programs

Mathematica

a[1] = 1; a[n_] := Times @@ ((#[[2]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 80}]
a[n_] := Sum[If[GCD[d, n/d] == 1, DivisorSigma[0, d], 0], {d, Divisors[n]}]; Table[a[n], {n, 80}]

PARI

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, numdiv(d))) \\ Andrew Howroyd, Apr 15 2021

PARI

for(n=1, 100, print1(direuler(p=2, n, (1 + X - X^2)/(1-X)^2)[n], ", ")) \\ Vaclav Kotesovec, Feb 11 2023

Python

from math import prod
from sympy import factorint
def A343443(n): return prod(e+2 for e in factorint(n).values()) # Chai Wah Wu, Feb 21 2025

Formula

a(n) = 2^omega(n) * tau_3(n) / tau(n), where omega = A001221, tau = A000005 and tau_3 = A007425.

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

From Bernard Schott, May 03 2021: (Start)

a(p^k) = k+2 for p prime, or signature [k].

a(A006881(n)) = 9 for signature [1, 1].

a(A054753(n)) = 12 for signature [2, 1].

a(A065036(n)) = 15 for signature [3, 1].

a(A085986(n)) = 16 for signature [2, 2].

a(A178739(n)) = 18 for signature [4, 1].

a(A143610(n)) = 20 for signature [3, 2].

a(A007304(n)) = 27 for signature [1, 1, 1]. (End)

Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 + 1/p^s - 1/p^(2*s)). - Vaclav Kotesovec, Feb 11 2023

From Amiram Eldar, Sep 01 2023: (Start)

a(n) = A000005(A064549(n)).

a(n) = A363194(A348018(n)). (End)