%I #9 Nov 11 2017 11:53:49
%S 1,2,6,9,20,26,43,63,75,91,130,151,185,243,279,307,383,392,488,511,
%T 595,716,904,917,1053,1213,1282,1262,1403,1632,1851,1839,1932,2135,
%U 2283,2426,2641,2913,3322,3347,3713,3642,4103,4386,4361,4893,5459
%N Number of integers k>=n such that binomial(k,n) has fewer than n distinct prime factors.
%C 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.
%H T. D. Noe, <a href="/A129233/b129233.txt">Table of n, a(n) for n=1..500</a>
%H Ernst S. Selmer, <a href="http://www.mscand.dk/article/viewFile/11662/9678">On the number of prime divisors of a binomial coefficient.</a> Math. Scand. 39 (1976), no. 2, 271-281.
%e 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.
%t 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}]]
%Y Cf. A005733, A005735.
%K nonn
%O 1,2
%A _T. D. Noe_, Apr 05 2007, May 20 2007