A385349 - OEIS

%I #21 Jun 28 2025 20:41:26

%S 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,

%T 1,1,33,17,35,27,1,19,39,5,1,441,1,11,2025,23,1,3,7,125,51,13,1,729,

%U 55,7,57,29,1,225,1,31,3969,1,65,1089,1,17,69,1225,1,27,1,37,5625

%N Product of odd proper divisors of n.

%H Alois P. Heinz, <a href="/A385349/b385349.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.

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

%p a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):

%p seq(a(n), n=1..75);  # _Alois P. Heinz_, Jun 27 2025

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

%o (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

%o (Python)

%o from math import isqrt

%o from sympy import divisor_count

%o def A385349(n):

%o     d = divisor_count(m:=n>>(~n&n-1).bit_length())

%o     k = isqrt(m)**d if d&1 else m**(d>>1)

%o     return k//n if n&1 else k # _Chai Wah Wu_, Jun 27 2025

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

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, Jun 26 2025