login
Product of the divisors d of n such that sqrt(n) < d < n.
1

%I #12 Jun 17 2023 11:25:31

%S 1,1,1,1,1,3,1,4,1,5,1,24,1,7,5,8,1,54,1,50,7,11,1,576,1,13,9,98,1,

%T 900,1,128,11,17,7,1944,1,19,13,1600,1,2058,1,242,135,23,1,36864,1,

%U 250,17,338,1,4374,11,3136,19,29,1,1080000,1,31,189,512,13,7986,1,578,23

%N Product of the divisors d of n such that sqrt(n) < d < n.

%F a(n) = Product_{d|n, sqrt(n) < d < n} d.

%F a(n) = A007956(n)/A072499(n).

%F a(n) = A007955(n)/(n*A072499(n)).

%e The divisors of 16 are {1,2,4,8,16} and the product of the divisors d of n such that sqrt(16) = 4 < d < 16 is 8, so a(16) = 8.

%e The divisors of 30 are {1,2,3,5,6,10,15,30} and the product of the divisors d of n such that sqrt(30) < d < 30 is 6*10*15 = 900, so a(30) = 900.

%t a[n_] := Product[If[n < d^2 < n^2, d, 1], {d, Divisors[n]}]; Array[a, 100] (* _Amiram Eldar_, Jun 08 2023 *)

%o (PARI) a(n) = vecprod(select(x->((sqrt(n)<x) && (x<n)), divisors(n))); \\ _Michel Marcus_, Jun 08 2023

%Y Cf. A007955, A007956, A072499, A363520.

%K nonn,easy

%O 1,6

%A _Wesley Ivan Hurt_, Jun 07 2023