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

1,4

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A157117 A322143 A264081 * A123602 A208896 A288972

Adjacent sequences: A061535 A061536 A061537 * A061539 A061540 A061541

Keyword

nonn

Author

Labos Elemer, May 15 2001

Extensions

Corrected and edited by Jaroslav Krizek, Mar 05 2009

Status

approved