A118742 Numbers n for which the expression n!/(n+1) is an integer.

0, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 24, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97

Offset

0,2

Comments

Also set of all n>=0, excluding 3, for which n+1 is composite. [Proof: (i) If n+1 is prime, there cannot be any factor in n! to cancel the n+1 in the denominator of the expression. (ii) If n+1=composite=a*b, a<b, consider the equivalent expression (n+1)!/(n+1)^2=1*2*..*a*..*b*..(a*b)/(a^2*b^2) in which factors obviously cancel. (iii) If n+1=square=a^2, a>2, (n+1)!/(n+1)^2 = 1*2*..*a*...*(2a)*..*a^2/a^4 in which factors also cancel.] - R. J. Mathar, Nov 22 2006

Links

Table of n, a(n) for n=0..71.

Formula

a(n) = A002808(n+1)-1 for n>=1. - R. J. Mathar, Nov 22 2006

Example

n=5 5!/(5+1)= 5*4*3*2*1/6 = 20.

Maple

P:=proc(n) local i, j; for i from 0 by 1 to n do j:=i!/(i+1); if trunc(j)=j then print(i); fi; od; end: P(200);

Mathematica

Select[Range[0, 100], IntegerQ[#!/(#+1)]&] (* Harvey P. Dale, Aug 24 2014 *)

Crossrefs

Cf. A060462, A120416, A270441.

Essentially the same as A072668.

Sequence in context: A241028 A171097 A166460 * A122904 A104693 A225648

Adjacent sequences: A118739 A118740 A118741 * A118743 A118744 A118745

Keyword

nonn

Author

Paolo P. Lava and Giorgio Balzarotti, May 22 2006

Extensions

Corrected (39 inserted) by Harvey P. Dale, Aug 24 2014

Status

approved