A136655 - OEIS

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

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OFFSET

1,3

COMMENTS

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

LINKS

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

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).

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 *)

PROG

(Haskell)

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

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

CROSSREFS

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

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

Cf. A125911, A126192.

Cf. A001227, A183063.