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A124000 Semiprimes in A006987(n), or semiprime binomial coefficients: C(n,k), 2 <= k <= n-2.

%I A124000 #20 Nov 05 2016 17:30:06
%S A124000 6,10,15,21,35,55,91,253,703,1081,1711,1891,2701,3403,5671,12403,
%T A124000 13861,15931,18721,25651,34453,38503,49141,60031,64261,73153,79003,
%U A124000 88831,104653,108811,114481,126253,146611,158203,171991,188191,218791,226801
%N A124000 Semiprimes in A006987(n), or semiprime binomial coefficients: C(n,k), 2 <= k <= n-2.
%C A124000 Conjecture: all a(n) except a(1) = 6 and a(2) = 10 are odd. Conjecture: all a(n) except a(5) = 35 are triangular numbers of the form p*(2p +/- 1) that belong to A068443(n) = {6, 10, 15, 21, 55, 91, 253, 703, 1081, 1711, 1891, 2701, ...} Triangular numbers with two distinct prime factors.
%C A124000 Besides 35 & 371953, all members were found by C(n, 2). - _Robert G. Wilson v_, Sep 16 2016
%C A124000 Of C(n,k), n: 4, 5, 6, 7, 11, 14, 23, 38, 47, 59, 62, 74, 83, 107, 158, 167, 179, 194, ..., . - _Robert G. Wilson v_, Sep 16 2016
%H A124000 Robert G. Wilson v, <a href="/A124000/b124000.txt">Table of n, a(n) for n = 1..11521</a>
%F A124000 Intersection of A001358 and A006987. - _Michael B. Porter_, Sep 17 2016
%e A124000 C(5,2) = 5!/(3!*2!) = 120/(6*2) = 10 is a semiprime (A001358), so 10 is in the sequence. - _Michael B. Porter_, Sep 17 2016
%t A124000 s = {}; Do[b = Binomial[n, k]; If[PrimeOmega@ b == 2, AppendTo[s, b]; Print@ b], {n, 3, 10000}, {k, 2, n/2}]; s (* _Robert G. Wilson v_, Nov 03 2006; updated Sep 16 2016 *)
%Y A124000 Cf. A006987, A095147, A068443, A000217, A005382, A005384, A001358.
%K A124000 nonn
%O A124000 1,1
%A A124000 _Alexander Adamchuk_, Oct 31 2006
%E A124000 More terms from _Robert G. Wilson v_, Nov 03 2006