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%I #4 Mar 30 2012 17:22:46
%S 0,0,0,0,1,0,1,2,1,0,1,1,1,1,0,1,3,2,1,2,3,2,2,1,3,0,2,0,3,3,1,3,4,1,
%T 2,2,2,3,3,1,3,3,2,3,4,2,1,4,2,4,4,2,2,5,3,2,1,2,1,6,1,4,4,0,4,3,3,2,
%U 4,3,4,6,3,3,6,3,5,6,2,5,5,1,4,5,4,2,4,3,3,5,2,5,7,3,4,4,3,4,4,3,4,7,3,4,8
%N The number of times that binomial(2n,n) has two prime factors that add to 2n.
%C In general, a(3n) is much greater than a(3n-1) and a(3n+1), which is apparent in the graph of this sequence.
%H T. D. Noe, <a href="/A127499/b127499.txt">Table of n, a(n) for n=1..10000</a>
%e Consider n=8. Then binomial(16,8)=12870, which has prime factors 2,3,5,11,13. There are two pairs of prime factors that sum to 16: (3,13) and (5,11). Hence a(8)=2.
%t Table[p=Rest[Transpose[FactorInteger[Binomial[2n,n]]][[1]]]; cnt=0; i=1; While[i<=Length[p] && p[[i]]<=n, If[MemberQ[p,2n-p[[i]]], cnt++ ]; i++ ]; cnt, {n,100}]
%Y Cf. A070542 (n such that a(n)=0).
%K nonn
%O 1,8
%A _T. D. Noe_, Jan 19 2007
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