A226833 Triangle whose n-th row has the smallest n semiprimes in an arithmetic progression.

4, 4, 6, 6, 10, 14, 10, 22, 34, 46, 10, 22, 34, 46, 58, 201, 205, 209, 213, 217, 221, 133, 185, 237, 289, 341, 393, 445, 133, 185, 237, 289, 341, 393, 445, 497, 635, 707, 779, 851, 923, 995, 1067, 1139, 1211, 697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561

Offset: 1

Author

T. D. Noe, Jun 28 2013

Keywords

• nonn
• tabl
• look

Comments

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.

Examples

			Triangle:
4,
4,   6,
6,   10,  14,
10,  22,  34,  46,
10,  22,  34,  46,  58,
201, 205, 209, 213, 217,  221,
133, 185, 237, 289, 341,  393,  445,
133, 185, 237, 289, 341,  393,  445,  497,
635, 707, 779, 851, 923,  995,  1067, 1139, 1211,
697, 793, 889, 985, 1081, 1177, 1273, 1369, 1465, 1561
		

Links

• T. D. Noe, Rows n = 1..32 of triangle, flattened
• Andrzej Nowicki, Second numbers in arithmetic progressions, arxiv 1306.6424

Crossrefs

Cf. A226834 (first term), A096003 (last term), A097824 (gaps).

Programs

Mathematica

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