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

%I A072513 #15 Sep 21 2018 02:12:46
%S A072513 1,1,2,6,4,60,6,168,48,360,10,47520,12,1092,1680,20160,16,440640,18,
%T A072513 820800,5040,4620,22,734469120,480,7800,11232,4953312,28,3946320000,
%U A072513 30,9999360,21120,17952,28560,439723468800,36,25308,35568,35852544000
%N A072513 Product of all n - d, where d < n and d is a divisor of n.
%H A072513 Ivan Neretin, <a href="/A072513/b072513.txt">Table of n, a(n) for n = 1..1000</a>
%F A072513 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).
%F A072513 a(p) = p-1, a(pq) = pq(p-1)(q-1)(pq-1), p and q prime.
%F A072513 If n is not a prime or the square of a prime then n divides a(n).
%e A072513 a(6) = (6-1)(6-2)(6-3) = 60.
%e A072513 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.
%t A072513 Table[Times @@ (n - Most[Divisors[n]]), {n, 1, 40}] (* _Ivan Neretin_, May 26 2015 *)
%o A072513 (PARI) for(n=1,40,d=divisors(n); print1(prod(j=1,matsize(d)[2]-1,n-d[j]),","))
%o A072513 (PARI) a(n)=factorback(apply(d->if(d<n,n-d,1),divisors(n))) \\ _Charles R Greathouse IV_, May 26 2015
%o A072513 (Haskell)
%o A072513 a072513 n = product $ map (n -) $ a027751_row n
%o A072513 -- _Reinhard Zumkeller_, May 27 2015
%Y A072513 Cf. A072512, A080497, A080498, A080500 (similar products), A258324 (LCM instead of product).
%Y A072513 Cf. A027751.
%K A072513 nonn
%O A072513 1,3
%A A072513 _Amarnath Murthy_, Jul 28 2002
%E A072513 Edited and extended by _Klaus Brockhaus_, Jul 31 2002