A376571 -id:A376571 - cp's OEIS frontend

A376569 Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the number of points of the form (m, prime(m)) aligned with the points (k, prime(k)) and (n, prime(n)) (where prime(k) denotes the k-th prime number).

2, 2, 3, 2, 3, 3, 2, 3, 4, 7, 2, 2, 2, 3, 2, 2, 2, 4, 2, 4, 8, 2, 3, 2, 3, 3, 3, 2, 2, 2, 4, 2, 4, 2, 4, 5, 2, 2, 2, 2, 3, 8, 8, 4, 6, 2, 2, 2, 2, 2, 2, 2, 5, 5, 2, 2, 2, 2, 2, 2, 8, 8, 2, 2, 8, 6, 2, 2, 2, 2, 2, 8, 8, 3, 3, 8, 5, 8, 2, 2, 2, 2, 2, 2, 2, 5, 5, 2, 5, 2, 2

Offset: 2

Rémy Sigrist, Sep 28 2024

			Table T(n, k) begins:
    2;
    2, 3;
    2, 3, 3;
    2, 3, 4, 7;
    2, 2, 2, 3, 2;
    2, 2, 4, 2, 4, 8;
    2, 3, 2, 3, 3, 3, 2;
    2, 2, 4, 2, 4, 2, 4, 5;
    2, 2, 2, 2, 3, 8, 8, 4, 6;
    2, 2, 2, 2, 2, 2, 2, 5, 5, 2;
    2, 2, 2, 2, 2, 8, 8, 2, 2, 8, 6;
    2, 2, 2, 2, 2, 8, 8, 3, 3, 8, 5, 8;
    ...

Rémy Sigrist, Table of n, a(n) for n = 2..10012 (rows for n = 2..142 flattened)

Cf. A376187, A376570, A376571.

T(n,k) = { my (x0 = k, y0 = prime(x0), x1 = n, y1 = prime(x1), s = (y1-y0)/(x1-x0), maxp = max(60184, exp(max(y0/x0, s) + 1.1)), x2 = 0, v = 0); forprime (y2 = 2, 1+maxp, x2++; if (x0 * (y1 - y2) + x1 * (y2 - y0) + x2 * (y0 - y1)==0, v++;);); return (v); }

T(n, k) >= 2.

A376570 Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the least m such that the points (m, prime(m)), (k, prime(k)) and (n, prime(n)) are aligned (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 3, 6, 1, 2, 3, 4, 2, 4, 7, 1, 2, 3, 4, 3, 6, 3, 8, 1, 2, 3, 4, 5, 6, 6, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 8, 10, 1, 2, 3, 4, 5, 6, 6, 8, 9, 6, 11, 1, 2, 3, 4, 5, 6, 6, 8, 9, 6, 11, 6, 1, 2, 3, 4, 5, 6, 7, 8, 8, 10, 8, 12, 13

Offset: 2

Rémy Sigrist, Sep 28 2024

			Triangle T(n, k) begins:
    1;
    1, 2;
    1, 2, 2;
    1, 2, 3, 4;
    1, 2, 3, 4, 5;
    1, 2, 3, 4, 3, 6;
    1, 2, 3, 4, 2, 4, 7;
    1, 2, 3, 4, 3, 6, 3, 8;
    1, 2, 3, 4, 5, 6, 6, 8, 9;
    1, 2, 3, 4, 5, 6, 7, 8, 8, 10;
    1, 2, 3, 4, 5, 6, 6, 8, 9, 6, 11;
    1, 2, 3, 4, 5, 6, 6, 8, 9, 6, 11, 6;
    ...

Rémy Sigrist, Table of n, a(n) for n = 2..10012 (rows for n = 2..142 flattened)
Rémy Sigrist, Scatterplot of (n, k) such that T(n, k) = k and n <= 1000

Cf. A376187, A376569, A376571.

T(n,k) = { my (x0 = k, y0 = prime(x0), x1 = n, y1 = prime(x1), x2 = 0); forprime (y2 = 2, oo, x2++; if (x0 * (y1 - y2) + x1 * (y2 - y0) + x2 * (y0 - y1)==0, return (x2););); }

T(n, k) <= k.

T(n, 1) = 1.

cp's OEIS Frontend

A376569 Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the number of points of the form (m, prime(m)) aligned with the points (k, prime(k)) and (n, prime(n)) (where prime(k) denotes the k-th prime number).

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