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%I #19 Jul 29 2019 11:50:49
%S 4,4,6,6,10,14,10,22,34,46,10,22,34,46,58,201,205,209,213,217,221,133,
%T 185,237,289,341,393,445,133,185,237,289,341,393,445,497,635,707,779,
%U 851,923,995,1067,1139,1211,697,793,889,985,1081,1177,1273,1369,1465,1561
%N Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.
%C The largest term of each row is as small as possible. Although Nowicki reports on the 28th row of this triangle, those terms are too large. Sequence A096003 reports the largest terms.
%H T. D. Noe, <a href="/A226833/b226833.txt">Rows n = 1..32 of triangle, flattened</a>
%H Andrzej Nowicki, <a href="http://arxiv.org/abs/1306.6424">Second numbers in arithmetic progressions</a>, arxiv 1306.6424
%e Triangle:
%e 4,
%e 4, 6,
%e 6, 10, 14,
%e 10, 22, 34, 46,
%e 10, 22, 34, 46, 58,
%e 201, 205, 209, 213, 217, 221,
%e 133, 185, 237, 289, 341, 393, 445,
%e 133, 185, 237, 289, 341, 393, 445, 497,
%e 635, 707, 779, 851, 923, 995, 1067, 1139, 1211,
%e 697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561
%t SemiPrimeQ[n_Integer] := If[Abs[n] < 2, False, (2 == Plus @@ Transpose[FactorInteger[Abs[n]]][[2]])]; p2 = Select[Range[2000], SemiPrimeQ]; nn = Length[p2]; t = {}; n = 0; last = 1; While[n++; found = False; last = n; While[k = last - 1; While[d = p2[[last]] - p2[[k]]; nums = Table[p2[[last]] - i*d, {i, 0, n - 1}]; int = Intersection[nums, Take[p2, last]]; nums[[-1]] > 0 && Length[int] < n, k--]; nums[[-1]] <= 0 && last < nn, last++]; If[last < nn, AppendTo[t, Reverse[nums]]]; last < nn]; t
%Y Cf. A226834 (first term), A096003 (last term), A097824 (gaps).
%K nonn,tabl,look
%O 1,1
%A _T. D. Noe_, Jun 28 2013