A350390 a(n) is the largest exponentially odd divisor of n.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A268335, A306071, A325837, A331737, A336643, A350389.

Sequence in context: A161871 A331737 A135875 * A366765 A361253 A097448

Adjacent sequences: A350387 A350388 A350389 * A350391 A350392 A350393

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 28 2021

Status

approved