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A385349 Product of odd proper divisors of n.

%I A385349 #21 Jun 28 2025 20:41:26
%S A385349 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 A385349 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 A385349 55,7,57,29,1,225,1,31,3969,1,65,1089,1,17,69,1225,1,27,1,37,5625
%N A385349 Product of odd proper divisors of n.
%H A385349 Alois P. Heinz, <a href="/A385349/b385349.txt">Table of n, a(n) for n = 1..10000</a>
%H A385349 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DivisorProduct.html">Divisor Product</a>.
%F A385349 a(n) = Product_{d|n, d < n, d odd} d.
%p A385349 a:= n-> mul(`if`(d::odd, d, 1), d=numtheory[divisors](n) minus {n}):
%p A385349 seq(a(n), n=1..75);  # _Alois P. Heinz_, Jun 27 2025
%t A385349 a[n_] := Times @@ Select[Divisors[n], # < n && OddQ[#] &]; Table[a[n], {n, 75}]
%o A385349 (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 A385349 (Python)
%o A385349 from math import isqrt
%o A385349 from sympy import divisor_count
%o A385349 def A385349(n):
%o A385349     d = divisor_count(m:=n>>(~n&n-1).bit_length())
%o A385349     k = isqrt(m)**d if d&1 else m**(d>>1)
%o A385349     return k//n if n&1 else k # _Chai Wah Wu_, Jun 27 2025
%Y A385349 Cf. A007955, A007956, A091570 (similar for sum), A136655, A385350 (fixed points).
%K A385349 nonn
%O A385349 1,6
%A A385349 _Ilya Gutkovskiy_, Jun 26 2025