A136655 Product of odd divisors of n.

1, 1, 3, 1, 5, 3, 7, 1, 27, 5, 11, 3, 13, 7, 225, 1, 17, 27, 19, 5, 441, 11, 23, 3, 125, 13, 729, 7, 29, 225, 31, 1, 1089, 17, 1225, 27, 37, 19, 1521, 5, 41, 441, 43, 11, 91125, 23, 47, 3, 343, 125, 2601, 13, 53, 729, 3025, 7, 3249, 29, 59, 225, 61, 31, 250047, 1, 4225, 1089

Offset: 1

Author

Jonathan Vos Post, Jun 25 2008

Keywords

• nonn
• easy

Comments

Product of rows of triangle A182469. - Reinhard Zumkeller, May 01 2012

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000265, A000593, A007955, A007956, A078701.

Cf. A140210, A140211, A140213, A140214, A140215.

Cf. A125911, A126192.

Cf. A001227, A183063.

Programs

Haskell

a136655 = product . a182469_row  -- Reinhard Zumkeller, May 01 2012

Maple

with(numtheory); f:=proc(n) local t1,i,k; t1:=divisors(n); k:=1; for i in t1 do if i mod 2 = 1 then k:=k*i; fi; od; k; end; # N. J. A. Sloane, Jul 14 2008

Mathematica

Array[Times @@ Select[Divisors@ #, OddQ] &, 66] (* Michael De Vlieger, Aug 03 2017 *)
a[n_] := (oddpart = n/2^IntegerExponent[n, 2])^(DivisorSigma[0, oddpart]/2); Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

PARI

a(n) = my(d=divisors(n)); prod(k=1, #d, if (d[k]%2, d[k], 1)); \\ Michel Marcus, Aug 04 2017

Python

from math import isqrt
from sympy import divisor_count
def A136655(n):
    d = divisor_count(m:=n>>(~n&n-1).bit_length())
    return isqrt(m)**d if d&1 else m**(d>>1) # Chai Wah Wu, Jun 27 2025

Formula

a(p) = p if p noncomposite; a(2^n) = 1; a(pq) = p^2 * q^2 when p, q are odd primes.

a(n) = sqrt(n^od(n)/2^ed(n)), where od(n) = number of odd divisors of n = tau(2*n)-tau(n) and ed(n) = number of even divisors of n = 2*tau(n)-tau(2*n). - Vladeta Jovovic, Jun 25 2008

Also a(n) = A007955(A000265(n)). - David Wilson, Jun 26 2008

a(n) = Product_{h == 1 mod 4 and h | n}*Product_{i == 3 mod 4 and i | n}.

a(n) = Product_{j == 1 mod 6 and j | n}*Product_{k == 5 mod 6 and k | n}.

a(n) = A140210(n)*A140211(n). - R. J. Mathar, Jun 27 2008

a(n) = A007955(n) / A125911(n).

Extensions

More terms from N. J. A. Sloane, Jul 14 2008

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar