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