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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).

%I A375421 #12 Feb 14 2025 08:25:47
%S A375421 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,
%T A375421 23,24,23,28,29,27,29,30,29,29,32,31,32,34,35,35,37,36,37,39,42,44,45,
%U A375421 45,43,44,47,47,48,48,49,46,49,49,49,56,56,53,55,61,62
%N 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).
%H A375421 Rémy Sigrist, <a href="/A375421/b375421.txt">Table of n, a(n) for n = 1..10000</a>
%F A375421 a(n) <= n-1.
%F A375421 a(n) = n-1-A334046(n). - _Pontus von Brömssen_, Feb 14 2025
%e A375421 The first terms, alongside the corresponding lines, are:  n  a(n)  Lines
%e A375421   -  ----  --------------------------------------------------------------
%e A375421   1     0  {}
%e A375421   2     1  {x + 1}
%e A375421   3     2  {2*x - 1, 3/2*x + 1/2}
%e A375421   4     2  {2*x - 1, 5/3*x + 1/3}
%e A375421   5     4  {4*x - 9, 3*x - 4, 8/3*x - 7/3, 9/4*x - 1/4}
%e A375421   6     5  {3*x - 5, 8/3*x - 3, 5/2*x - 2, 2*x + 1, 11/5*x - 1/5}
%e A375421   7     5  {4*x - 11, 3*x - 4, 10/3*x - 19/3, 14/5*x - 13/5, 5/2*x - 1/2}
%o A375421 (PARI) { for (n = 1, #p = primes(68), print1 (#Set(vector(n-1, i, polinterpolate([i, n], [p[i], p[n]])))", ");); }
%Y A375421 Cf. A334046, A375420.
%K A375421 nonn
%O A375421 1,3
%A A375421 _Rémy Sigrist_, Aug 14 2024