2, 3, 7, 23, 47, 181, 83, 73, 1069, 521, 701, 1627, 691, 4271, 4261, 3733, 3943, 3929, 10369, 509, 10463, 24683, 10259, 4297, 4159, 34963, 4021, 157907, 24923, 24691, 4027, 162007, 26759, 27283, 164821, 164503, 187721, 164839, 27067, 180437, 27143, 27059, 164663, 27043, 189961
Offset: 1
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.
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