A376187 For a line L in the plane, let C(L) denote the number of prime points [k, prime(k)] on L, and let M(L) denote the maximum prime(k) for any of these points; a(n) = minimum M(L) over all lines with C(L) = n, or -1 if there is no such line.

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

Author

N. J. A. Sloane, Sep 23 2024

Keywords

• nonn
• more

Comments

C(L) is the total number of prime points on L, by definition.

This sequence minimizes the largest prime in any line containing n prime-points. For the maximal smallest prime in any line that has the minimal largest prime (i.e. the lines arising in the present sequence), see A376190.

If C(L) = n in the definition is changed to C(L) >= n we get A376188.

Other known values are a(47) = 189887, a(48) = 164707, a(50)-a(58) = [180511, 180463, 26947, 193373, 180289, 180541, 164627, 194083, 186311], a(60) = 193871, a(62)-a(65) = [187471, 194239, 194309, 194141], a(67)-a(70) = [194269, 193723, 193513, 192737], a(76)-a(79) = [194069, 194267, 193789, 193841]. - Max Alekseyev, Sep 27 2024.

Examples

			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.
		

Links

• W. Edwin Clark, A line of slope 6 containing 20 prime-points (blue dots), and a parallel line, also with 20 prime-points (red dots)
• N. J. A. Sloane, Table of lines in the plane containing the known maximum numbers of prime-points
• N. J. A. Sloane, Sketch taken from A373813 which includes lines corresponding to a(3) = 7 and a(4) = 23
• N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Crossrefs

Cf. A005115, A373813, A376188, A376190.

Extensions

a(9) corrected by Rémy Sigrist, Sep 24 2024.

a(12) from W. Edwin Clark, Sep 25 2024.

a(14)-a(45) from Max Alekseyev, Sep 26 2024, and independently confirmed by Don Reble, Oct 02 2024.