A093000 Least k such that Sum_{r=n+1..k} r >= n!.

2, 3, 5, 8, 16, 38, 101, 284, 852, 2694, 8935, 30952, 111598, 417560, 1617204, 6468816, 26671611, 113158064, 493244565, 2205856753, 10108505545, 47413093714, 227385209453, 1113955476429, 5569777382146, 28400403557929

Offset: 1

Author

Amarnath Murthy, Mar 29 2004

Keywords

• easy
• nonn

Comments

Equivalently, least k such that the product of the first n positive integers is less than the sum of the integers from n+1 through k.

a(n) = floor(sqrt(2*n! + n^2)) for most values of n; the exceptions are 1,2,3,7,..., in which case a(n) = floor(sqrt(2*n! + n^2)) + 1.

Examples

			a(4) = 8 because 4! = 24 and 5+6+7+8 = 26 > 24, but 5+6+7 = 18.
a(5) = 16 because 5! = 120 and 6+7+8+...+15+16 = 121 > 120.
		

Crossrefs

Cf. A093001.

Programs

PARI

{ for(n=1,20, s=0; found=0; for(k=n+1,10000000, if( k*(k+1)-n*(n+1)>= 2*n!, print1(k,","); found=1; break; ); ); if(found==0, print(0); ); ); } \\ R. J. Mathar, Apr 21 2006

Formula

Least k such that {k(k+1)/2 - n(n+1)/2} >= n!.

a(n) = ceiling((-1 + sqrt(1 + 8n! + 4n^2 + 4n))/2) and ignoring the -1 outside the sqrt and the 1 inside gives the approximate formula in the comment. - Joshua Zucker, May 08 2006

Extensions

More terms from R. J. Mathar, Apr 21 2006

More terms from Joshua Zucker, May 08 2006

Name simplified by Jon E. Schoenfield, Jun 15 2019