A375421 -id:A375421 - cp's OEIS frontend

A334046 a(n) is the number of points of the set { (k, prime(k)), k = 1..n-1 } that are not visible from the point (n, prime(n)) (where prime(n) denotes the n-th prime number).

0, 0, 0, 1, 0, 0, 1, 2, 2, 1, 1, 2, 3, 2, 4, 4, 3, 5, 2, 3, 6, 5, 6, 4, 2, 3, 5, 4, 4, 6, 2, 2, 5, 4, 4, 6, 7, 5, 7, 7, 6, 6, 7, 6, 8, 8, 7, 5, 4, 4, 5, 8, 8, 6, 7, 7, 8, 8, 12, 10, 11, 12, 6, 7, 11, 10, 5, 5, 7, 6, 10, 10, 10, 11, 11, 12, 15, 12, 13, 13, 11

Offset: 1

Rémy Sigrist, Apr 13 2020

nonn

For any i and k such that i < k: the point (i, prime(i)) is visible from the point (k, prime(k)) if there are no j such that i < j < k and the three points (i, prime(i)), (j, prime(j)), (k, prime(k)) are aligned.

			For n = 4:
- we consider the following points:
     .   .   .   X
                / (4,7)
     .   .   . / .
              /
     .   .   X   .
            / (3,5)
     .   . / .   .
          /
     .   X   .   .
          (2,3)
     X   .   .   .
      (1,2)
     .   .   .   .
- (2,3) is not visible from (4,7) (as it is hidden by (3,5)),
- whereas (1,2) and (3,5) are visible from (4,7),
- hence a(4) = 1.

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

Cf. A000040, A334043, A375421.

g(z) = z/gcd(real(z), imag(z))
for (n=1, 81, print1 (n-1-#Set(apply(k -> g((k+prime(k)*I)-(n+prime(n)*I)), [1..n-1]))", "))

a(n) = n-1-A375421(n). - Pontus von Brömssen, Feb 14 2025

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).

Original entry on oeis.org

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

Rémy Sigrist, Aug 14 2024

nonn

			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}

Cf. A000217, A375421.

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

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

cp's OEIS Frontend

A334046 a(n) is the number of points of the set { (k, prime(k)), k = 1..n-1 } that are not visible from the point (n, prime(n)) (where prime(n) denotes the n-th prime number).

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