A375420 a(n) is the number of straight lines passing through the points (i, prime(i)) and (j, prime(j)) with i < j <= n (where prime(k) denotes the k-th prime number).

0, 1, 3, 4, 8, 13, 17, 20, 25, 32, 40, 48, 56, 66, 74, 84, 94, 105, 120, 135, 148, 162, 176, 191, 211, 230, 247, 267, 287, 306, 333, 361, 385, 410, 437, 462, 488, 517, 545, 571, 601, 633, 663, 696, 730, 761, 796, 836, 879, 921, 962, 1000, 1038, 1080, 1122

Offset: 1

Author

Rémy Sigrist, Aug 14 2024

Keywords

• nonn

Examples

			The first terms, alongside the new lines passing through (n, prime(n)), are:
  n  a(n)  New lines passing through (n, prime(n))
  -  ----  --------------------------------------------
  1     0  {}
  2     1  {x + 1}
  3     3  {2*x - 1, 3/2*x + 1/2}
  4     4  {5/3*x + 1/3}
  5     8  {4*x - 9, 3*x - 4, 8/3*x - 7/3, 9/4*x - 1/4}
		

Crossrefs

Cf. A000217, A375421.

Programs

PARI

{ L = []; for (n = 1, #p = primes(55), print1 (#L = setunion(L, Set(vector(n-1, i, polinterpolate([i, n], [p[i], p[n]]))))", ");); }

Formula

a(n) <= A000217(n+1).