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A376497 a(n) is the maximum number of points from the set {(prime(k), prime(k+1)), k = 1..n} belonging to a straight line passing through the point (prime(n), prime(n+1)) (where prime(k) denotes the k-th prime number).

%I A376497 #6 Sep 28 2024 06:22:26
%S A376497 1,2,2,2,3,2,4,3,3,5,2,4,6,5,3,4,7,5,6,8,6,7,7,3,8,9,9,10,10,3,11,8,
%T A376497 11,3,12,9,10,12,11,12,13,3,14,13,15,3,2,14,16,15,13,17,3,14,15,16,18,
%U A376497 17,16,19,4,4,17,20,18,3,18,5,21,19,19,3,20,21,20
%N A376497 a(n) is the maximum number of points from the set {(prime(k), prime(k+1)), k = 1..n} belonging to a straight line passing through the point (prime(n), prime(n+1)) (where prime(k) denotes the k-th prime number).
%C A376497 Is this sequence unbounded?
%H A376497 Rémy Sigrist, <a href="/A376497/b376497.txt">Table of n, a(n) for n = 1..10000</a>
%H A376497 Rémy Sigrist, <a href="/A376497/a376497.gp.txt">PARI program</a>
%e A376497 The first terms, alongside an appropriate set of points, are:
%e A376497   n   a(n)  Points
%e A376497   --  ----  ------------------------------------------------
%e A376497    1     1  (2,3)
%e A376497    2     2  (2,3), (3,5)
%e A376497    3     2  (3,5), (5,7)
%e A376497    4     2  (5,7), (7,11)
%e A376497    5     3  (3,5), (5,7), (11,13)
%e A376497    6     2  (7,11), (13,17)
%e A376497    7     4  (3,5), (5,7), (11,13), (17,19)
%e A376497    8     3  (7,11), (13,17), (19,23)
%e A376497    9     3  (3,5), (13,17), (23,29)
%e A376497   10     5  (3,5), (5,7), (11,13), (17,19), (29,31)
%e A376497   11     2  (23,29), (31,37)
%e A376497   12     4  (7,11), (13,17), (19,23), (37,41)
%e A376497   13     6  (3,5), (5,7), (11,13), (17,19), (29,31), (41,43)
%e A376497   14     5  (7,11), (13,17), (19,23), (37,41), (43,47)
%e A376497   15     3  (23,29), (31,37), (47,53)
%e A376497   16     4  (23,29), (31,37), (47,53), (53,59)
%o A376497 (PARI) \\ See Links section.
%Y A376497 Cf. A375422.
%K A376497 nonn
%O A376497 1,2
%A A376497 _Rémy Sigrist_, Sep 25 2024