This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle begins:
2;
2, 3;
3, 5, 7;
5, 11, 17, 23;
5, 11, 17, 23, 29;
7, 37, 67, 97, 127, 157;
7, 157, 307, 457, 607, 757, 907;
199, 409, 619, 829, 1039, 1249, 1459, 1669;
199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879;
199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089;
...
A005115(7) comes from the 7-term prime progression {7, 157, 307, 457, 607, 757, 907}, and so 7 is in this sequence. - _Charlie Neder_, Feb 02 2019
a = {{ 1, 2, 2}, {2, 2 + j, 3}, {3, 3 + 2j, 7}, {4, 5 + 6j, 23}, {5, 5 + 6j, 29}, {6, 7 + 30j, 157}, {7, 7 + 150j, 907}, {8, 199 + 210j, 1669}, {9, 199 + 210j, 1879}, {10, 199 + 210j, 2089}, {11, 110437 + 13860j, 249037}, {12, 110437 + 13860j, 262897}}
Union[Table[CoefficientList[a[[n, 2]], j][[1]], {n, 1, 12}]]
First and smallest occurrence of n, n >= 1, consecutive primes in arithmetic progression: a(1) = 2: (2) (degenerate arithmetic progression); a(2) = 2: (2, 3) (degenerate arithmetic progression); a(3) = 3: (3, 5, 7); a(4) = 251: (251, 257, 263, 269); a(5) = 9843019: (9843019, 9843049, 9843079, 9843109, 9843139); a(6) = 121174811: (121174811, 121174841, 121174871, 121174901, 121174931, 121174961);
Join[{2},Table[SelectFirst[Partition[Prime[Range[691*10^4]],n,1], Length[ Union[ Differences[ #]]] == 1&][[1]],{n,2,6}]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 10 2019 *)
f:= proc(n) local d,k; d:= 2*n; for k from 1 while isprime(19+d*k) do od: k end proc: map(f, [$1..200]); # Robert Israel, Jul 27 2020
bb={}; Do[s=1; Do[If[PrimeQ[19+k*d], s=s+1, bb={bb, s}; Break[]], {k, 10}], {d, 2, 200, 2}]; Flatten[bb]
The following are lines corresponding to a(1) to a(8). We describe the lines by simply listing the primes "prime(k)" corresponding to the points on the line. n L 1 2 2 2,3 3 3,5,7 4 5,11,17,23 5 19,23,31,43,47 6 61,71,101,131,151,181 7 7,11,59,67,71,79,83 8 13,17,29,37,41,53,61,73 There are two parallel lines of slope 6 which both contain 20 points. The first contains the points with [x,y] coordinates [45, 197], [51, 233], [52, 239], [54, 251], [55, 257], [56, 263], [57, 269], [64, 311], [71, 353], [72, 359], [76, 383], [77, 389], [79, 401], [86, 443], [87, 449], [89, 461], [92, 479], [94, 491], [96, 503], [97, 509] (here y == -1 mod 6), and the second contains the points [42, 181], [44, 193], [47, 211], [50, 229], [63, 307], [67, 331], [68,337], [70, 349], [73, 367], [74, 373], [75, 379], [78, 397], [80, 409],[82, 421], [84, 433], [85, 439], [88, 457], [93, 487], [95, 499], [99, 523] (here y == 1 mod 6). The existence of these two lines was confirmed by _W. Edwin Clark_, who produced the illustration in the LINKS section. This shows an enlargement of the region 35 <= x <= 105. The blue dots are the points on the first line, the red dots those on the second line. It is interesting to contrast these two 20-point lines with the results in A005115, which gives the earliest arithmetic progressions of primes with a given number of terms. To find an arithmetic progression of 20 primes one has to go out to 572945039351. Of course these primes don't lie on a line, because of the irregular spacing between the primes. For many further examples of lines containing many prime-points see the Table in the LINKS section. There are at least five lines of 54 points each and slope 12; and at least 56 lines of 18 points each and slope 12. There is a 79-point line, connecting (12125,129533)-(17484,193841), again with slope 12. Populous slope-12 lines are common within my search range. - _Don Reble_, Oct 02 2024.
a(3)=14 since (4,9,14) [or (6,10,14)] is an arithmetic progression of 3 semiprimes ending in 14 and 14 is the smallest semiprime with this property.
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