A390812 a(n) is the number of pairs (i, j) with 1 <= i < j < n such that the prime-points (i, prime(i)), (j, prime(j)), and (n, prime(n)) are collinear.

0, 0, 0, 1, 0, 0, 1, 2, 3, 1, 1, 3, 6, 3, 7, 10, 3, 15, 3, 6, 21, 11, 16, 4, 2, 3, 6, 5, 4, 8, 3, 3, 8, 4, 5, 9, 14, 8, 17, 8, 8, 12, 10, 9, 24, 10, 11, 11, 10, 5, 6, 11, 11, 7, 10, 13, 18, 11, 16, 13, 13, 16, 9, 22, 16, 11, 11, 15, 17, 21, 31, 38, 31, 40, 47, 48, 60, 57, 68, 68, 26

Offset

1,8

Comments

Equivalently, i and j must have the same slope (prime(n) - prime(i))/(n - i). Hence, a(n) is the sum of the number of ways one can choose two among all indices that have a given slope s, over all possible values s, see Formula. - M. F. Hasler, Apr 12 2026

With prime(111), it is the first time that prime(1) gets involved, as (prime(111) - prime(1))/(111 - 1) == (prime(111) - prime(103))/(111 - 103) == 11/2. - Ruud H.G. van Tol, Apr 23 2026

Links

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

Formula

a(n) = Sum_{s in S} binomial(m(s), 2), with S = { slope(i); 0 < i < n }, slope(i) = (prime(n) - prime(i))/(n - i) and m(s) = #{ i < n : slope(i) = s }. - M. F. Hasler, Apr 12 2026

Example

a(4) = 1 because the only set of triples in this case is {(2, 3), (3, 5), (4, 7)}: these three are on a line of slope 2.
a(100) = 4: the four triples are {(27, 103), (31, 127), (100, 541)}, {(60, 281), (72, 359), (100, 541)}, {(79, 401), (82, 421), (100, 541)}, {(86, 443), (88, 457), (100, 541)}.

Prog

(PARI) isok(v1, v2, v3) = (v3[2] - v2[2])/(v3[1] - v2[1]) == (v2[2] - v1[2])/(v2[1] - v1[1]);
a(n) = my(pn=prime(n)); sum(i=1, n-1, my(pi=prime(i)); sum(j=i+1, n-1, isok([i, pi], [j, prime(j)], [n, pn]))); \\ Michel Marcus, Apr 10 2026
(PARI) apply( {A390812(n)=my(p=primes(n), L); sum(i=1, #p=vecsort([(p[n]-p[i])/(n-i)|i<-[1..n-1]]), if(i+1==n||p[i]!=p[i+1], binomial(-L+L=i, 2)))}, [1..99]) \\ M. F. Hasler, Apr 12 2026
(PARI) A390812(n)=if(n>1, my(M=Map(), p=primes(n), f); for(i=1, n-1, mapput(M, f=(p[n]-p[i])/(n-i), iferr(mapget(M, f), E, )+1)); vecsum([t*(t-1)\2|t<-Mat(M)[, 2], t>1])) \\ M. F. Hasler, Apr 12 2026

Crossrefs

Cf. A393445, A373813, A393446.

Sequence in context: A265848 A139438 A377842 * A135392 A071947 A139343

Adjacent sequences: A390809 A390810 A390811 * A390813 A390814 A390815

Keyword

nonn

Author

Firdous Ahmad Mala, Apr 09 2026

Status

approved