A061538 Product of all divisors of n, divided by product of unitary divisors; or equivalently product of non-unitary divisors of n.

1, 1, 1, 2, 1, 1, 1, 8, 3, 1, 1, 12, 1, 1, 1, 64, 1, 18, 1, 20, 1, 1, 1, 576, 5, 1, 27, 28, 1, 1, 1, 1024, 1, 1, 1, 7776, 1, 1, 1, 1600, 1, 1, 1, 44, 45, 1, 1, 110592, 7, 50, 1, 52, 1, 2916, 1, 3136, 1, 1, 1, 3600, 1, 1, 63, 32768, 1, 1, 1, 68, 1, 1, 1, 26873856, 1, 1, 75, 76, 1, 1, 1

Offset: 1

Author

Labos Elemer, May 15 2001

Keywords

• nonn

Examples

			For n = 16: only {1,16} are unitary, while {2,4,8} are non-unitary divisors, so a(16) = 64.
When all divisors are unitary, then A048105 is 0 and the corresponding terms here are equal to 1.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000005, A007955, A007956, A034444, A048105.

Programs

Mathematica

Table[Times @@ Select[Divisors@ n, ! CoprimeQ[#, n/#] &], {n, 79}] (* Michael De Vlieger, Mar 17 2017 *)
a[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Array[a, 80] (* Amiram Eldar, Jul 22 2024 *)

PARI

{ for (n=1, 1000, s=divisors(n); a=1; for (i=2, length(s), d=s[i]; if (gcd(d, n/d)!=1, a*=d)); write("b061538.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 24 2009

PARI

a(n) = {my(f = factor(n)); n^((numdiv(f) - 2^omega(f))/2);} \\ Amiram Eldar, Jul 22 2024

Formula

a(n) = n^(A048105(n)/2) = n^((A000005(n) - A034444(n))/2).

Extensions

Corrected and edited by Jaroslav Krizek, Mar 05 2009