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a(n) is the maximum number of points from the set {(k, prime(k)), k = 1..n} belonging to a straight line passing through the point (n, prime(n)) (where prime(k) denotes the k-th prime number).
5

%I #12 Aug 17 2024 12:20:36

%S 1,2,2,3,2,2,3,3,4,3,3,4,5,4,5,6,3,7,4,5,8,6,7,3,3,3,4,4,3,4,4,4,5,3,

%T 4,5,6,5,7,4,4,6,5,5,8,4,5,6,6,4,4,4,5,4,5,6,7,5,4,5,4,5,5,8,5,4,6,7,

%U 7,8,9,10,9,10,11,11,12,12,13,13,8,14,9,15

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

%C Is this sequence unbounded?

%H Rémy Sigrist, <a href="/A375422/b375422.txt">Table of n, a(n) for n = 1..10000</a>

%H Rémy Sigrist, <a href="/A375422/a375422.txt">C++ program</a>

%H Rémy Sigrist, <a href="/A375422/a375422.gp.txt">PARI program</a>

%e The first terms, alongside an appropriate set of points, are:

%e n a(n) Points

%e -- ---- --------------------------------------------------

%e 1 1 (1,2)

%e 2 2 (1,2), (2,3)

%e 3 2 (1,2), (3,5)

%e 4 3 (2,3), (3,5), (4,7)

%e 5 2 (1,2), (5,11)

%e 6 2 (1,2), (6,13)

%e 7 3 (3,5), (5,11), (7,17)

%e 8 3 (2,3), (5,11), (8,19)

%e 9 4 (3,5), (5,11), (7,17), (9,23)

%e 10 3 (6,13), (7,17), (10,29)

%e 11 3 (8,19), (9,23), (11,31)

%e 12 4 (6,13), (7,17), (10,29), (12,37)

%e 13 5 (6,13), (7,17), (10,29), (12,37), (13,41)

%e 14 4 (8,19), (9,23), (11,31), (14,43)

%e 15 5 (8,19), (9,23), (11,31), (14,43), (15,47)

%e 16 6 (6,13), (7,17), (10,29), (12,37), (13,41), (16,53)

%o (C++) // See Links section.

%o (PARI) \\ See Links section.

%Y Cf. A334046, A375386.

%K nonn

%O 1,2

%A _Rémy Sigrist_, Aug 14 2024