A124570 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.

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, 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, 4, 6, 6, 0, 1333, 0, 481, 0, 0, 0, 4, 4, 4, 77, 69, 0, 1333, 0, 5989, 0, 0, 0, 4

Offset

1,1

Comments

Comment from Hugo van der Sanden Aug 14 2021: (Start)

Row d=12 starts 4 9 9 10 10 469 3937 7343 7343 44719 78937 78937 78937 78937 55952333 233761133 597191343199.

Row d=18 starts 4 4 15 15 15 695 695 1727 7711 13951 13951 46159 400847 400847 400847 65737811 13388955301 934046384293.

Row d=24 starts 4 9 9 10 10 793 4819 6415 7271 14069 14069 14069 31589 67344271 616851797 48299373047 48299373047 20302675273219.

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.

Row d=36 starts 4 10 10 10 15 1333 3161 4997 6865 34885 142171 834863 1327447 35528747 720945097 63389173477 63389173477 16074207679897 41728758250241.

Row d=42 starts 4 4 9 35 35 2701 2987 2987 7729 26995 26995 185795 307553 708385 708385 708385 1090198367 1819546069 20263042201 5672249016001.

Later terms in these rows are always >10^14. (End)

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

Links

Table of n, a(n) for n=1..78.

R. J. Mathar, Table for d <= 999 (PDF)

Formula

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

Example

Array begins:
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.
0..|..4...4...4...4...4....4....4....4......4....4.....4.....4...
1..|..4...9...33..0...0....0....0....0......0....0.....0.....0....
2..|..4...4...91..213.213..1383.3091.8129...0....0.....0.....0.....
3..|..4...6...115.0...0....0....0....0......0....0.....0.....0.....
4..|..4...6...6...111.201..201..481..5989...0....0.....0.....0....
5..|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
6..|..4...4...9...203.1333.1333.1333.2159...8309.18799.60499.60499
7..|..4...14..51..0...0....0....0....0......0....0.....0.....0.....
8..|..4...6...6...69..473..511..511..112697.0....0.....0.....0.....
9..|..4...6...77..0...0....0....0....0......0....0.....0.....0.....
10.|..4...4...15..289.289..289..1631.13501..0....0.....0.....0.....
11.|..4...4...4...0...0....0....0....0......0....0.....0.....0.....
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.

Crossrefs

Semiprime analog of A124064.

Cf. A125025 (row lengths), A001358, A056809, A070552, A092125, A092126, A092127, A092128, A092129, A124064, A092209 (row d=2), A091016 (row d=6).

Sequence in context: A081676 A114555 A361471 * A213083 A053187 A013189

Adjacent sequences: A124567 A124568 A124569 * A124571 A124572 A124573

Keyword

nonn,tabl,changed

Author

Jonathan Vos Post, Nov 04 2006

Extensions

Corrected and extended by R. J. Mathar, Nov 06 2006

Definition clarified by Robert Israel, Aug 11 2024

Status

approved