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A365331
The number of divisors of the largest square dividing n.
3
1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 3, 3, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 9, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 5, 3, 3, 1, 3, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 3, 7, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 5, 5, 1, 1, 3, 1, 1, 1
OFFSET
1,4
COMMENTS
All the terms are odd.
The sum of these divisors is A365332(n).
The number of divisors of the square root of the largest square dividing n is A046951(n).
LINKS
FORMULA
a(n) = A000005(A008833(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = e + 1 - (e mod 2).
Dirichlet g.f.: zeta(s)*zeta(2*s)^2/zeta(4*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5/2.
More precise asymptotics: Sum_{k=1..n} a(k) ~ 5*n/2 + 3*zeta(1/2)*sqrt(n)/Pi^2 * (log(n) + 4*gamma - 2 - 24*zeta'(2)/Pi^2 + zeta'(1/2)/zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 02 2023
MAPLE
a:= n-> mul(2*iquo(i[2], 2)+1, i=ifactors(n)[2]):
seq(a(n), n=1..100); # Alois P. Heinz, Sep 01 2023
MATHEMATICA
f[p_, e_] := e + 1 - Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> x + 1 - x%2, factor(n)[, 2]));
(PARI) a(n) = numdiv(n/core(n)); \\ Michel Marcus, Sep 02 2023
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 01 2023
STATUS
approved