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

1,2

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.

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.

Example

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.

Mathematica

Join[{1}, Table[cnt=1; n=k; b=1; n0=Infinity; While[n++; b=b*n/(n-k); If[Length[FactorInteger[b]]<k, cnt=cnt+1; n0=n]; n<10*n0]; cnt, {k, 2, 20}]]

Crossrefs

Cf. A005733, A005735.

Sequence in context: A182984 A301798 A093840 * A106529 A325040 A350949

Adjacent sequences: A129230 A129231 A129232 * A129234 A129235 A129236

Keyword

nonn

Author

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

Status

approved