A385349 Product of odd proper divisors of n.

1, 1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 3, 1, 7, 15, 1, 1, 27, 1, 5, 21, 11, 1, 3, 5, 13, 27, 7, 1, 225, 1, 1, 33, 17, 35, 27, 1, 19, 39, 5, 1, 441, 1, 11, 2025, 23, 1, 3, 7, 125, 51, 13, 1, 729, 55, 7, 57, 29, 1, 225, 1, 31, 3969, 1, 65, 1089, 1, 17, 69, 1225, 1, 27, 1, 37, 5625

Offset: 1

Author

Ilya Gutkovskiy, Jun 26 2025

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Divisor Product.

Crossrefs

Cf. A007955, A007956, A091570 (similar for sum), A136655, A385350 (fixed points).

Programs

Maple

a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):
seq(a(n), n=1..75);  # Alois P. Heinz, Jun 27 2025

Mathematica

a[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Table[a[n], {n, 75}]

PARI

a(n) = my(m = n >> valuation(n,2), d = numdiv(m)); if(d % 2, sqrtint(m)^d, m^(d/2)) / if(m < n, 1, n); \\ Amiram Eldar, Jun 27 2025

Python

from math import isqrt
from sympy import divisor_count
def A385349(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    k = isqrt(m)**d if d&1 else m**(d>>1)
    return k//n if n&1 else k # Chai Wah Wu, Jun 27 2025

Formula

a(n) = Product_{d|n, d < n, d odd} d.