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