A061538 - OEIS

%I #18 Jul 22 2024 15:23:24

%S 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,

%T 1,1024,1,1,1,7776,1,1,1,1600,1,1,1,44,45,1,1,110592,7,50,1,52,1,2916,

%U 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

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

%H Harry J. Smith, <a href="/A061538/b061538.txt">Table of n, a(n) for n = 1..1000</a>

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

%e For n = 16: only {1,16} are unitary, while {2,4,8} are non-unitary divisors, so a(16) = 64.

%e When all divisors are unitary, then A048105 is 0 and the corresponding terms here are equal to 1.

%t Table[Times @@ Select[Divisors@ n, ! CoprimeQ[#, n/#] &], {n, 79}] (* _Michael De Vlieger_, Mar 17 2017 *)

%t a[n_] := n^((DivisorSigma[0, n] - 2^PrimeNu[n]) / 2); Array[a, 80] (* _Amiram Eldar_, Jul 22 2024 *)

%o (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

%o (PARI) a(n) = {my(f = factor(n)); n^((numdiv(f) - 2^omega(f))/2);} \\ _Amiram Eldar_, Jul 22 2024

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

%K nonn

%O 1,4

%A _Labos Elemer_, May 15 2001

%E Corrected and edited by _Jaroslav Krizek_, Mar 05 2009