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A365335
The number of exponentially odd coreful divisors of the square root of the largest square dividing n.
2
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,64
COMMENTS
First differs from A160338 at n = 64, and from A178489 at n = 65.
The number of divisors of the square root of the largest square dividing n is A046951(n).
The number of exponentially odd divisors of the square root of the largest square dividing n is A365549(n) and their sum is A365336(n). [corrected, Sep 08 2023]
LINKS
FORMULA
a(n) = A325837(A000188(n)).
a(n) > 1 if and only if n is a bicubeful number (A355265).
Multiplicative with a(p^e) = floor((e+2)/4).
Dirichlet g.f.: zeta(s) * zeta(4*s) * Product_{p prime} (1 - 1/p^(4*s) + 1/p^(6*s)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 - 1/p^4 + 1/p^6) = 1.0181534831085... .
MATHEMATICA
f[p_, e_] := Max[1, Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> max(1, (x+2)\4), factor(n)[, 2]));
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 01 2023
EXTENSIONS
Name corrected by Amiram Eldar, Sep 08 2023
STATUS
approved