A109892 a(n) = least integer of the form (n!+1)(n!+2)...(n!+k)/n!.

2, 6, 84, 20475, 234531275, 199200973555045, 16481425431663122588749, 173392935733620216899469862542865, 300717095810709134168380432250652303057474577

Offset: 1

Author

Amarnath Murthy, Jul 13 2005

Keywords

• nonn

Comments

Equivalently, binomial(n!+n,n). Proof: (n!+1)(n!+2)...(n!+k) == k! mod n! == 0 mod n! if and only if k >= n (for n >= 2). - Paul D Hanna and Robert Israel, Aug 31 2010.

Note that k <= n. Subsidiary sequence to be investigated: n such that k < n.

This is just a coincidence, but k=2,6,84 are also such that floor(exp(1)*10^k) is a prime, cf. A064118. - M. F. Hasler, Aug 31 2013

Examples

			a(4)=25*26*27*28/24=20475.
		

Crossrefs

Cf. A105291.

Programs

Maple

A109892 := proc(n) local k,fn; k := 1; fn := n! ; while mul(fn+i,i=1..k) mod fn <> 0 do k := k+1; od ; RETURN(mul(fn+i,i=1..k)/fn) ; end: seq(A109892(n),n=1..10) ; # R. J. Mathar, Aug 15 2007

Mathematica

Table[(n+n!)!/(n!*(n!)!), {n, 1, 9}] (* Jean-François Alcover, Mar 04 2014, after first comment *)

Extensions

Corrected and extended by R. J. Mathar, Aug 15 2007