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

1,6

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Divisor Product.

Formula

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

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}]

Prog

(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

Crossrefs

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

Sequence in context: A075001 A252840 A093803 * A285175 A016599 A079650

Adjacent sequences: A385346 A385347 A385348 * A385350 A385351 A385352

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 26 2025

Status

approved