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

%I A129233 #9 Nov 11 2017 11:53:49
%S A129233 1,2,6,9,20,26,43,63,75,91,130,151,185,243,279,307,383,392,488,511,
%T A129233 595,716,904,917,1053,1213,1282,1262,1403,1632,1851,1839,1932,2135,
%U A129233 2283,2426,2641,2913,3322,3347,3713,3642,4103,4386,4361,4893,5459
%N A129233 Number of integers k>=n such that binomial(k,n) has fewer than n distinct prime factors.
%C A129233 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 A129233 T. D. Noe, <a href="/A129233/b129233.txt">Table of n, a(n) for n=1..500</a>
%H A129233 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 A129233 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 A129233 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 A129233 Cf. A005733, A005735.
%K A129233 nonn
%O A129233 1,2
%A A129233 _T. D. Noe_, Apr 05 2007, May 20 2007