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

1,3

Links

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

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).

Example

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.

Mathematica

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

Prog

(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(d<n, n-d, 1), divisors(n))) \\ Charles R Greathouse IV, May 26 2015
(Haskell)
a072513 n = product $ map (n -) $ a027751_row n
-- Reinhard Zumkeller, May 27 2015

Crossrefs

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

Cf. A027751.

Sequence in context: A330078 A258324 A080499 * A022404 A210420 A306585

Adjacent sequences: A072510 A072511 A072512 * A072514 A072515 A072516

Keyword

nonn

Author

Amarnath Murthy, Jul 28 2002

Extensions

Edited and extended by Klaus Brockhaus, Jul 31 2002

Status

approved