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A350390
a(n) is the largest exponentially odd divisor of n.
15
1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 8, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 24, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 32, 65, 66, 67, 34, 69
OFFSET
1,2
COMMENTS
First differs from A331737 at n = 16.
LINKS
FORMULA
Multiplicative with a(p^e) = p^e if e is odd and p^(e-1) otherwise.
a(n) = n/A336643(n).
a(n) = n if and only if n is an exponentially odd number (A268335).
Sum_{k=1..n} a(k) ~ (1/2)*c*n^2, where c = Product_{p prime} 1-(p-1)/(p^2*(p+1)) = 0.8073308216... (A306071).
Dirichlet g.f.: zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) + 1/p^(2*s-1)). - Amiram Eldar, Sep 18 2023
MATHEMATICA
f[p_, e_] := If[OddQ[e], p^e, p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(Python)
from math import prod
from sympy.ntheory.factor_ import primefactors, core
def A350390(n): return n*core(n)//prod(primefactors(n)) # Chai Wah Wu, Dec 30 2021
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(f[i, 2] - !(f[i, 2]%2))); } \\ Amiram Eldar, Sep 18 2023
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 28 2021
STATUS
approved