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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).

%I A375420 #9 Aug 18 2024 20:20:31
%S A375420 0,1,3,4,8,13,17,20,25,32,40,48,56,66,74,84,94,105,120,135,148,162,
%T A375420 176,191,211,230,247,267,287,306,333,361,385,410,437,462,488,517,545,
%U A375420 571,601,633,663,696,730,761,796,836,879,921,962,1000,1038,1080,1122
%N 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).
%F A375420 a(n) <= A000217(n+1).
%e A375420 The first terms, alongside the new lines passing through (n, prime(n)), are:
%e A375420   n  a(n)  New lines passing through (n, prime(n))
%e A375420   -  ----  --------------------------------------------
%e A375420   1     0  {}
%e A375420   2     1  {x + 1}
%e A375420   3     3  {2*x - 1, 3/2*x + 1/2}
%e A375420   4     4  {5/3*x + 1/3}
%e A375420   5     8  {4*x - 9, 3*x - 4, 8/3*x - 7/3, 9/4*x - 1/4}
%o A375420 (PARI) { L = []; for (n = 1, #p = primes(55), print1 (#L = setunion(L, Set(vector(n-1, i, polinterpolate([i, n], [p[i], p[n]]))))", ");); }
%Y A375420 Cf. A000217, A375421.
%K A375420 nonn
%O A375420 1,3
%A A375420 _Rémy Sigrist_, Aug 14 2024