cp's OEIS Frontend

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.

%I A376187 #97 Jul 09 2025 05:06:14
%S A376187 2,3,7,23,47,181,83,73,1069,521,701,1627,691,4271,4261,3733,3943,3929,
%T A376187 10369,509,10463,24683,10259,4297,4159,34963,4021,157907,24923,24691,
%U A376187 4027,162007,26759,27283,164821,164503,187721,164839,27067,180437,27143,27059,164663,27043,189961
%N 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.
%C A376187 C(L) is the total number of prime points on L, by definition.
%C A376187 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.
%C A376187 If C(L) = n in the definition is changed to C(L) >= n we get A376188.
%C A376187 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.
%H A376187 W. Edwin Clark, <a href="/A376187/a376187.pdf">A line of slope 6 containing 20 prime-points (blue dots), and a parallel line, also with 20 prime-points (red dots) </a>
%H A376187 N. J. A. Sloane, <a href="/A376187/a376187_3.txt">Table of lines in the plane containing the known maximum numbers of prime-points</a>
%H A376187 N. J. A. Sloane, <a href="/A373813/a373813.pdf">Sketch taken from A373813 which includes lines corresponding to a(3) = 7 and a(4) = 23</a>
%H A376187 N. J. A. Sloane, <a href="https://www.youtube.com/watch?v=3RAYoaKMckM">A Nasty Surprise in a Sequence and Other OEIS Stories</a>, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/sloane85BD.pdf">Slides</a> [Mentions this sequence]
%e A376187 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.
%e A376187   n   L
%e A376187   1   2
%e A376187   2   2,3
%e A376187   3   3,5,7
%e A376187   4   5,11,17,23
%e A376187   5   19,23,31,43,47
%e A376187   6   61,71,101,131,151,181
%e A376187   7   7,11,59,67,71,79,83
%e A376187   8   13,17,29,37,41,53,61,73
%e A376187 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),
%e A376187 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).
%e A376187 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.
%e A376187 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.
%e A376187 For many further examples of lines containing many prime-points see the Table in the LINKS section.
%e A376187 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.
%Y A376187 Cf. A005115, A373813, A376188, A376190.
%K A376187 nonn,more
%O A376187 1,1
%A A376187 _N. J. A. Sloane_, Sep 23 2024
%E A376187 a(9) corrected by _Rémy Sigrist_, Sep 24 2024.
%E A376187 a(12) from _W. Edwin Clark_, Sep 25 2024.
%E A376187 a(14)-a(45) from _Max Alekseyev_, Sep 26 2024, and independently confirmed by _Don Reble_, Oct 02 2024.