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A325837
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The number of coreful divisors of n that are exponentially odd numbers (A268335).
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11
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1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
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OFFSET
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1,8
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COMMENTS
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First differs from A050361 at n = 64.
The number of exponentially odd divisors of n is A322483(n), and their sum is A033634(n).
A coreful divisor d of a number n is a divisor with the same set of distinct prime factors as n. (End)
Also, the number of divisors of n that are cubefull exponentially odd numbers (A335988). - Amiram Eldar, Feb 11 2024
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LINKS
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FORMULA
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Multiplicative with a(p^e) = floor((e+1)/2).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.231291... (A065487). - Amiram Eldar, Sep 10 2022
a(n) = A046951(A350390(n)) (the number of squares dividing the largest exponentially odd divisor of n). - Amiram Eldar, Sep 01 2023
Dirichlet g.f.: zeta(s) * zeta(2*s) * Product_{p prime} (1 - 1/p^(2*s) + 1/p^(3*s)). (End)
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MATHEMATICA
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fun[p_, e_] := Floor[(e+1)/2]; a[n_] := Times@@(fun@@@FactorInteger[n]); Array[a, 100]
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PROG
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(PARI) a(n) = vecprod(apply(x -> (x+1)\2, factor(n)[, 2])); \\ Amiram Eldar, Sep 01 2023
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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