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A113691 Semiprimes in A033951.

%I A113691 #8 Sep 08 2022 08:45:23
%S A113691 46,77,218,1073,1351,1502,1661,2186,2998,4193,4727,5006,5293,5891,
%T A113691 7183,8603,10558,12266,13631,14581,15563,19811,20953,25202,27806,
%U A113691 29843,30538,31241,32671,33398,35627,37153,39502,40301,46118,46981,49618,56051
%N A113691 Semiprimes in A033951.
%C A113691 This sequence, A113691, contains semiprimes from the center straight down the y-axis in the semiprime spiral of A113688-A113689. A113693 contains semiprimes from the center straight up the y-axis in the semiprime spiral. A113690 contains semiprimes from the center straight right along the x-axis in the semiprime spiral. Semiprimes from the center straight left along the x-axis in the semiprime spiral are A113692.
%H A113691 Vincenzo Librandi, <a href="/A113691/b113691.txt">Table of n, a(n) for n = 1..1000</a>
%F A113691 {a(n)} = Intersection of A001358 and A033951. Semiprimes of the form 4*k^2 + 3*k + 1.
%e A113691 a(5) = 4*18^2 + 3*18 + 1 = 1351 = 7 * 193.
%e A113691 a(6) = 4*19^2 + 3*19 + 1 = 1502 = 2 * 751.
%e A113691 a(7) = 4*20^2 + 3*20 + 1 = 1661 = 11 * 151.
%e A113691 a(5), a(6) and a(7) are vertically adjacent in the semiprime spiral, hence part of a clump and not isolated semiprimes as in A113688. a(11), a(12) and a(13) are another such vertical string of 3 adjacent semiprimes and so is a(26), a(27) and a(28).
%e A113691 a(52) = 4*152^2 + 3*152 + 1 = 92873 = 11 * 8443 is the greatest member under 10^5.
%t A113691 Select[Table[4*n^2 + 3*n + 1, {n, 200}], PrimeOmega[#] == 2&] (* _Vincenzo Librandi_, Sep 22 2012 *)
%o A113691 (Magma) IsSemiprime:= func<n | &+[d[2]: d in Factorization(n)] eq 2>; [s: n in [1..120] | IsSemiprime(s) where s is 4*n^2 + 3*n + 1]; // _Vincenzo Librandi_, Sep 22 2012
%Y A113691 Cf. A001358, A033951, A113688-A113699.
%K A113691 easy,nonn
%O A113691 1,1
%A A113691 _Jonathan Vos Post_, Nov 05 2005