A061537 Product of unitary divisors of n.

1, 2, 3, 4, 5, 36, 7, 8, 9, 100, 11, 144, 13, 196, 225, 16, 17, 324, 19, 400, 441, 484, 23, 576, 25, 676, 27, 784, 29, 810000, 31, 32, 1089, 1156, 1225, 1296, 37, 1444, 1521, 1600, 41, 3111696, 43, 1936, 2025, 2116, 47, 2304, 49, 2500, 2601, 2704, 53, 2916

Offset: 1

Author

Labos Elemer, May 15 2001

Keywords

• nonn

Comments

Also appears to be smallest number m such that A066296(m) = n.

Examples

			For n = 288, unitary divisors = {1, 9, 32, 288}, a(288) = 1 * 9 * 32 * 288 = 82944.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000 (terms n=1..1000 from Harry J. Smith)

Crossrefs

Cf. A000005, A001221, A007955, A007956, A034444, A048105, A066296.

Programs

Maple

a:= n-> mul(`if`(igcd(d, n/d)=1, d, 1), d=numtheory[divisors](n)):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 01 2017

Mathematica

Table[Times@@ Select[Divisors[n], GCD[#, n/#]==1 &], {n, 1, 100}] (* Indranil Ghosh, Aug 04 2017 *)
a[n_] := n^(2^(PrimeNu[n]-1)); Array[a, 60] (* 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("b061537.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 24 2009

PARI

a(n) = n^(2^(omega(n)-1)); \\ Amiram Eldar, Jul 22 2024

Python

from sympy import divisors, gcd, prod
def a(n): return prod(d for d in divisors(n) if gcd(d, n//d)==1)
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 04 2017

Formula

a(n) = n^(A034444(n)/2) = n^(2^(A001221(n)-1)).

Extensions

Corrected and edited by Jaroslav Krizek, Mar 05 2009