A129233 Number of integers k>=n such that binomial(k,n) has fewer than n distinct prime factors.

1, 2, 6, 9, 20, 26, 43, 63, 75, 91, 130, 151, 185, 243, 279, 307, 383, 392, 488, 511, 595, 716, 904, 917, 1053, 1213, 1282, 1262, 1403, 1632, 1851, 1839, 1932, 2135, 2283, 2426, 2641, 2913, 3322, 3347, 3713, 3642, 4103, 4386, 4361, 4893, 5459

Offset: 1

Author

T. D. Noe, Apr 05 2007, May 20 2007

Keywords

• nonn

Comments

This sequence, which is much smoother than the closely related A005735, is calculated using the same "cheat" as described in Selmer's paper. That is, after we seem to have found the largest k for a given n, we search up to 10k for binomial coefficients having fewer than n distinct prime factors.

Examples

			Consider n=3. The values of binomial(k,n) are 1,4,10,20,35,56,84,120 for k=3..10. Selmer shows that k=8 yields the largest value having fewer than 3 distinct prime factors. Factoring the other values shows that a(3)=6.
		

Links

• T. D. Noe, Table of n, a(n) for n=1..500
• Ernst S. Selmer, On the number of prime divisors of a binomial coefficient. Math. Scand. 39 (1976), no. 2, 271-281.

Crossrefs

Cf. A005733, A005735.

Programs

Mathematica

Join[{1}, Table[cnt=1; n=k; b=1; n0=Infinity; While[n++; b=b*n/(n-k); If[Length[FactorInteger[b]]