A376569 -id:A376569 - cp's OEIS frontend

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).

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.

A376571 Table T(n, k), n > 1, k = 1..n-1, read by rows; T(n, k) is the greatest 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

2, 3, 4, 4, 4, 4, 5, 8, 9, 23, 6, 6, 6, 8, 6, 7, 7, 9, 7, 9, 21, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 15, 10, 10, 10, 10, 15, 21, 21, 52, 152, 11, 11, 11, 11, 11, 11, 11, 15, 15, 11, 12, 12, 12, 12, 12, 21, 21, 12, 12, 21, 153, 13, 13, 13, 13, 13, 21, 21, 28, 17, 21, 53, 21

Offset: 2

Rémy Sigrist, Sep 28 2024

			Table T(n, k) begins:
    2;
    3, 4;
    4, 4, 4;
    5, 8, 9, 23;
    6, 6, 6, 8, 6;
    7, 7, 9, 7, 9, 21;
    8, 8, 8, 8, 8, 8, 8;
    9, 9, 9, 9, 9, 9, 9, 15;
    10, 10, 10, 10, 15, 21, 21, 52, 152;
    11, 11, 11, 11, 11, 11, 11, 15, 15, 11;
    12, 12, 12, 12, 12, 21, 21, 12, 12, 21, 153;
    13, 13, 13, 13, 13, 21, 21, 28, 17, 21, 53, 21;
    ...

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

Cf. A376187, A376569, A376570.

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 = -oo); forprime (y2 = 2, 1+maxp, x2++; if (x0 * (y1 - y2) + x1 * (y2 - y0) + x2 * (y0 - y1)==0, v = x2;);); return (v); }

T(n, k) >= n.

cp's OEIS Frontend

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).

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