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A365333
The number of exponentially odd coreful divisors of the largest square dividing n.
2
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, 2, 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, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
OFFSET
1,16
COMMENTS
First differs from A043289, A053164, A063775, A203640 and A295658 at n = 64.
The number of squares dividing the largest exponentially odd divisor of n is A325837(n).
The sum of the exponentially odd divisors of the largest square dividing n is A365334(n). [corrected, Sep 08 2023]
The number of exponentially odd divisors of the largest square dividing n is the same as the number of squares dividing n, A046951(n). - Amiram Eldar, Sep 08 2023
LINKS
FORMULA
a(n) = A325837(A008833(n)).
a(n) = 1 if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = max(1, floor(e/2)).
Dirichlet g.f.: zeta(s) * zeta(4*s) * zeta(6*s) / zeta(12*s).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 15015/(1382*Pi^2) = 1.100823... .
MATHEMATICA
f[p_, e_] := Max[1, Floor[e/2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = vecprod(apply(x -> max(1, x\2), factor(n)[, 2]));
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Sep 01 2023
EXTENSIONS
Name corrected by Amiram Eldar, Sep 08 2023
STATUS
approved