A350389 a(n) is the largest unitary divisor of n that is an exponentially odd number (A268335).

1, 2, 3, 1, 5, 6, 7, 8, 1, 10, 11, 3, 13, 14, 15, 1, 17, 2, 19, 5, 21, 22, 23, 24, 1, 26, 27, 7, 29, 30, 31, 32, 33, 34, 35, 1, 37, 38, 39, 40, 41, 42, 43, 11, 5, 46, 47, 3, 1, 2, 51, 13, 53, 54, 55, 56, 57, 58, 59, 15, 61, 62, 7, 1, 65, 66, 67, 17, 69, 70, 71

Offset

1,2

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 1 otherwise.

a(n) = n/A350388(n).

A001222(a(n)) = A350387(n).

a(n) = 1 if and only if n is a positive square (A000290 \ {0}).

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+p^2+p^3)) = 0.7406196365...

Dirichlet g.f.: zeta(2*s-2) * zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) - 1/p^(2*s-2) - 1/p^(3*s-1)). - Amiram Eldar, Sep 18 2023

Mathematica

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

Prog

(Python)
from math import prod
from sympy import factorint
def A350389(n): return prod(p**e if e % 2 else 1 for p, e in factorint(n).items()) # Chai Wah Wu, Feb 24 2022
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^f[i, 2], 1)); } \\ Amiram Eldar, Sep 18 2023

Crossrefs

Cf. A000290, A001222, A007913, A268335, A350387, A350388, A350390.

Sequence in context: A065883 A214392 A071975 * A368886 A182659 A359593

Adjacent sequences: A350386 A350387 A350388 * A350390 A350391 A350392

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 28 2021

Status

approved