A072513 Product of all n - d, where d < n and d is a divisor of n.

1, 1, 2, 6, 4, 60, 6, 168, 48, 360, 10, 47520, 12, 1092, 1680, 20160, 16, 440640, 18, 820800, 5040, 4620, 22, 734469120, 480, 7800, 11232, 4953312, 28, 3946320000, 30, 9999360, 21120, 17952, 28560, 439723468800, 36, 25308, 35568, 35852544000

Offset: 1

Author

Amarnath Murthy, Jul 28 2002

Keywords

• nonn

Examples

			a(6) = (6-1)(6-2)(6-3) = 60.
For n = 16 the divisors d < n are 1,2,4 and 8, so a(16) = (16-1)*(16-2)*(16-4)*(16-8) = 15*14*12*8 = 20160.
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A072512, A080497, A080498, A080500 (similar products), A258324 (LCM instead of product).

Cf. A027751.

Programs

Haskell

a072513 n = product $ map (n -) $ a027751_row n
-- Reinhard Zumkeller, May 27 2015

Mathematica

Table[Times @@ (n - Most[Divisors[n]]), {n, 1, 40}] (* Ivan Neretin, May 26 2015 *)

PARI

for(n=1,40,d=divisors(n); print1(prod(j=1,matsize(d)[2]-1,n-d[j]),","))

PARI

a(n)=factorback(apply(d->if(dCharles R Greathouse IV, May 26 2015

Formula

a(n) = (n-d_1)(n-d_2)...(n-d_k) where d_k is the largest divisor of n less than n (k = tau(n) - 1).

a(p) = p-1, a(pq) = pq(p-1)(q-1)(pq-1), p and q prime.

If n is not a prime or the square of a prime then n divides a(n).

Extensions

Edited and extended by Klaus Brockhaus, Jul 31 2002