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A357669
a(n) is the number of divisors of the powerful part of n.
5
1, 1, 1, 3, 1, 1, 1, 4, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 4, 3, 1, 4, 3, 1, 1, 1, 6, 1, 1, 1, 9, 1, 1, 1, 4, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1
OFFSET
1,4
COMMENTS
The corresponding sum of divisors of the powerful part of n is A295294.
LINKS
FORMULA
a(n) = A000005(A057521(n)).
a(n) = A000005(n)/A056671(n).
a(n) = A000005(A064549(A003557(n))).
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000005(n) iff n is powerful (A001694).
Multiplicative with a(p^e) = 1 if e = 1 and e+1 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 - p^2 + 2*p - 1)/(p^2*(p - 1))) = 2.71098009471568319328... .
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^s + 2/p^(2*s) - 1/p^(3*s)). - Amiram Eldar, Sep 09 2023
MATHEMATICA
f[p_, e_] := If[e == 1, 1, e + 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(e = factor(n)[, 2]); prod(i=1, #e, if(e[i] == 1, 1, e[i] + 1))};
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Oct 08 2022
STATUS
approved