A068333 Product(n/k - k) where the product is over the divisors k of n and where 1 <= k <= sqrt(n).

0, 1, 2, 0, 4, 5, 6, 14, 0, 27, 10, 44, 12, 65, 28, 0, 16, 357, 18, 152, 80, 189, 22, 2300, 0, 275, 156, 972, 28, 2639, 30, 1736, 256, 495, 68, 0, 36, 629, 380, 12636, 40, 8569, 42, 6020, 2112, 945, 46, 215072, 0, 5635, 700, 11016, 52, 59625

Offset: 1

Author

Leroy Quet, Feb 27 2002

Keywords

• nonn
• look

Comments

From Robert Israel, Jun 02 2019: (Start)

a(n) is divisible by n-1.

a(n) = 0 if and only if n is a square.

a(n) = n-1 if n is prime. (End)

Examples

			a(8) = (8 - 1) (4 - 2) = 14 because 1 and 2 are the divisors of 8 which are <= sqrt(8).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Programs

Maple

f:= proc(n) local D,k;
  D:= select(t -> t^2 <= n, numtheory:-divisors(n));
  mul(n/k-k, k=D)
end proc:
map(f, [$1..100]); # Robert Israel, Jun 02 2019

Mathematica

a[n_] := Product[If[1 <= k <= Sqrt[n], (n/k - k), 1], {k, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Aug 16 2020 *)

PARI

a(n) = my(p=1); fordiv(n, d, if (d^2 <= n, p *= n/d - d)); p; \\ Michel Marcus, Jun 02 2019