A375421 a(n) is the number of distinct straight lines passing through the points (k, prime(k)) and (n, prime(n)) for k < n (where prime(k) denotes the k-th prime number).

0, 1, 2, 2, 4, 5, 5, 5, 6, 8, 9, 9, 9, 11, 10, 11, 13, 12, 16, 16, 14, 16, 16, 19, 22, 22, 21, 23, 24, 23, 28, 29, 27, 29, 30, 29, 29, 32, 31, 32, 34, 35, 35, 37, 36, 37, 39, 42, 44, 45, 45, 43, 44, 47, 47, 48, 48, 49, 46, 49, 49, 49, 56, 56, 53, 55, 61, 62

Offset: 1

Author

Rémy Sigrist, Aug 14 2024

Keywords

• nonn

Examples

			The first terms, alongside the corresponding lines, are:  n  a(n)  Lines
  -  ----  --------------------------------------------------------------
  1     0  {}
  2     1  {x + 1}
  3     2  {2*x - 1, 3/2*x + 1/2}
  4     2  {2*x - 1, 5/3*x + 1/3}
  5     4  {4*x - 9, 3*x - 4, 8/3*x - 7/3, 9/4*x - 1/4}
  6     5  {3*x - 5, 8/3*x - 3, 5/2*x - 2, 2*x + 1, 11/5*x - 1/5}
  7     5  {4*x - 11, 3*x - 4, 10/3*x - 19/3, 14/5*x - 13/5, 5/2*x - 1/2}
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A334046, A375420.

Programs

PARI

{ for (n = 1, #p = primes(68), print1 (#Set(vector(n-1, i, polinterpolate([i, n], [p[i], p[n]])))", ");); }

Formula

a(n) <= n-1.

a(n) = n-1-A334046(n). - Pontus von Brömssen, Feb 14 2025