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%I #34 Aug 12 2024 01:22:52
%S 4,4,4,4,9,4,4,4,33,4,4,6,91,0,4,4,6,115,213,0,4,4,4,6,0,213,0,4,4,4,
%T 4,111,0,1383,0,4,4,14,9,0,201,0,3091,0,4,4,6,51,203,0,201,0,8129,0,4,
%U 4,6,6,0,1333,0,481,0,0,0,4,4,4,77,69,0,1333,0,5989,0,0,0,4
%N Array read by antidiagonals: T(d,k) (k >= 1, d = 1,2,3,4,5,6,...) = smallest semiprime s of k (not necessarily consecutive) semiprimes in arithmetic progression with common difference d, or 0 if there is no such arithmetic progression.
%C Comment from _Hugo van der Sanden_ Aug 14 2021: (Start)
%C Row d=12 starts 4 9 9 10 10 469 3937 7343 7343 44719 78937 78937 78937 78937 55952333 233761133 597191343199.
%C Row d=18 starts 4 4 15 15 15 695 695 1727 7711 13951 13951 46159 400847 400847 400847 65737811 13388955301 934046384293.
%C Row d=24 starts 4 9 9 10 10 793 4819 6415 7271 14069 14069 14069 31589 67344271 616851797 48299373047 48299373047 20302675273219.
%C Row d=30 starts 4 4 9 25 25 2779 2779 6347 6347 6347 10811 10811 87109 87109 87109 1513723 15009191 15009191 316612697 316612697 1275591688621.
%C Row d=36 starts 4 10 10 10 15 1333 3161 4997 6865 34885 142171 834863 1327447 35528747 720945097 63389173477 63389173477 16074207679897 41728758250241.
%C Row d=42 starts 4 4 9 35 35 2701 2987 2987 7729 26995 26995 185795 307553 708385 708385 708385 1090198367 1819546069 20263042201 5672249016001.
%C Later terms in these rows are always >10^14. (End)
%C If p is the least prime that does not divide d, then T(d,k) <= p^2 if k >= p^2 (i.e. any a.p. of length >= p^2 with difference d contains a term divisible by p^2, and the only semiprime divisible by p^2 is p^2). Thus every row is eventually 0. - _Robert Israel_, Aug 11 2024
%H R. J. Mathar, <a href="/A124570/a124570_1.pdf">Table for d <= 999</a> (PDF)
%F T(1,2)=A070552(1). T(1,3)=A056809(1). T(2,4)=A092126(1). T(2,5)=A092127(1). T(2,6)=A092128(1). T(2,7)=A092129(1). T(2,8)=A082919(1). T(3,2)=A123017(1). T(d,1)=A001358(1). - _R. J. Mathar_, Aug 05 2021
%e Array begins:
%e d.\...k=1.k=2.k=3.k=4.k=5..k=6..k=7..k=8....k=9..k=10.k=11..k=12.
%e 0..|..4...4...4...4...4....4....4....4......4....4.....4.....4...
%e 1..|..4...9...33..0...0....0....0....0......0....0.....0.....0....
%e 2..|..4...4...91..213.213..1383.3091.8129...0....0.....0.....0.....
%e 3..|..4...6...115.0...0....0....0....0......0....0.....0.....0.....
%e 4..|..4...6...6...111.201..201..481..5989...0....0.....0.....0....
%e 5..|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
%e 6..|..4...4...9...203.1333.1333.1333.2159...8309.18799.60499.60499
%e 7..|..4...14..51..0...0....0....0....0......0....0.....0.....0.....
%e 8..|..4...6...6...69..473..511..511..112697.0....0.....0.....0.....
%e 9..|..4...6...77..0...0....0....0....0......0....0.....0.....0.....
%e 10.|..4...4...15..289.289..289..1631.13501..0....0.....0.....0.....
%e 11.|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
%e Example for row 3: 115 = 5 * 23 is semiprime, 115+3 = 118 = 2 * 59 is semiprime and 115+3+3 = 121 = 11^2 is semiprime, so T(3,3) = 115.
%Y Semiprime analog of A124064.
%Y Cf. A125025 (row lengths), A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129, A124064, A092209 (row d=2), A091016 (row d=6).
%K nonn,tabl,changed
%O 1,1
%A _Jonathan Vos Post_, Nov 04 2006
%E Corrected and extended by _R. J. Mathar_, Nov 06 2006
%E Definition clarified by _Robert Israel_, Aug 11 2024
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