A370329 a(n) is the number of coreful divisors of the n-th powerful number that are also powerful numbers.

1, 1, 2, 1, 3, 1, 2, 4, 1, 1, 5, 2, 3, 1, 2, 1, 2, 6, 3, 1, 1, 2, 4, 1, 4, 7, 4, 1, 3, 2, 1, 2, 3, 6, 1, 1, 2, 8, 1, 5, 3, 6, 2, 1, 5, 3, 4, 1, 8, 1, 1, 2, 4, 4, 9, 1, 2, 6, 1, 1, 9, 2, 2, 2, 1, 2, 1, 1, 4, 5, 1, 10, 1, 2, 1, 3, 8, 6, 3, 10, 1, 6, 2, 1, 7, 2, 3

Offset: 1

Author

Amiram Eldar, Feb 15 2024

Keywords

• nonn
• easy

Comments

A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n (see A307958).

The positive terms of A361430.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001694, A002117, A062503, A078434, A307958, A360908 (analogous with squares), A361430, A370328.

Programs

Mathematica

f[p_, e_] := e - 1; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; With[{max = 10^4}, s /@ Union@ Flatten@ Table[i^2*j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}]]

PARI

lista(kmax) = {my(e); for(k = 1, kmax, e = factor(k)[,2]; if(k == 1 || vecmin(e) > 1, print1(prod(i = 1, #e, e[i]-1), ", ")));}

Formula

a(n) = A361430(A001694(n)).

a(n) = 1 if and only if n is the square of a squarefree number (A062503).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(3/2) * zeta(3) * Product_{p prime} (1 + 2/p^2 + 2/p^(5/2) - 1/p^3 - 2/p^(7/2) - 2/p^4) = 6.91748056612108993003... . (The infinite product of primes is the value of f(1/2) in A361430).