

A093000


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


1



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
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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: A306622 A030034 A308852 * A122630 A108054 A123612
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



