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

1,1

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.

Links

Table of n, a(n) for n=1..26.

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

Example

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.

Prog

(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

Crossrefs

Cf. A093001.

Sequence in context: A030034 A370002 A308852 * A122630 A108054 A342690

Adjacent sequences: A092997 A092998 A092999 * A093001 A093002 A093003

Keyword

easy,nonn

Author

Amarnath Murthy, Mar 29 2004

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

Status

approved