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

Author

Amiram Eldar, Dec 28 2021

Keywords

• nonn
• easy
• mult

Comments

First differs from A331737 at n = 16.

Links

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

Crossrefs

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

Programs

Mathematica

f[p_, e_] := If[OddQ[e], p^e, p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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

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

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