A376187 -id:A376187 - cp's OEIS frontend

A376190 For a line L in the plane, let C(L) denote the number of prime points [k, prime(k)] on L, and let M(L) denote the maximum prime(k) for any of these points. a(n) is the maximum of the smallest primes in the lines L with C(L) = n and containing prime A376187(n), or a(n) = -1 if no such lines exist.

2, 2, 3, 5, 19, 18, 7, 13, 967, 113, 83, 619, 103, 1583, 1693, 1621, 1759, 1753, 5923, 197, 6143, 15823, 5849, 1609, 1663, 10333, 1613, 152003, 15683, 16111, 1619, 141871, 15649, 15383, 140989, 141811, 136481, 141319, 15667, 136769, 16033, 16619, 141707, 15473, 135649

Offset: 1

N. J. A. Sloane, Sep 25 2024, following a suggestion from W. Edwin Clark

Consider all the lines L in the plane containing exactly n prime-points (k, prime(k)). A376187 minimizes the maximal prime on any such line L, while the present sequence then maximizes the minimal prime on the lines from A376187.
In other words, we first minimize (in A376187) the maximal prime over all lines with exactly n points, and then here we further maximize the minimal prime. The second step minimizes the spread of the points.
For most listed terms, there is only one line L with C(L) = n that contains prime A376187(n). - Max Alekseyev, Sep 28 2024

			The best line with 5 points contains the primes 19,23,31,43,47, so a(5) = 19 and A376187(5) = 47. See the Table for further examples.

N. J. A. Sloane, Table of lines in the plane containing the known maximum numbers of prime-points

Cf. A376187, A376188.

Better definition and a(28)-a(45) from Max Alekseyev, Sep 28 2024

A373813 a(n) is the smallest number of straight lines needed to intersect all points (k, prime(k)) for k = 1..n.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15

Offset: 1

Rémy Sigrist and N. J. A. Sloane, Aug 18 2024

nonn

Dan Asimov asks if the graph is trying to converge to the Cantor (or Devil's Staircase) function. - N. J. A. Sloane, Aug 25 2024

Max Alekseyev, Table of n, a(n) for n = 1..410
Max Alekseyev, Sage program for lines covering points, Github, Aug 19 2024
N. J. A. Sloane, Sketch to illustrate first 11 terms. Solutions (representing points by their X-coordinates): a(5)=2: {1,5}{2,3,4}; a(9)=3: {1,2}{3,5,7,9}{4,6,8}; a(11)=4: {1,5}{2,3,4}{6,7,10}{8,9,11}.
N. J. A. Sloane, A Nasty Surprise in a Sequence and Other OEIS Stories, Experimental Mathematics Seminar, Rutgers University, Oct 10 2024, Youtube video; Slides [Mentions this sequence]

Cf. A373814 (run lengths), A373810 (same with y(k) = phi(k)), A373811 (similar with y(k) = a(k)), A375499 (same with y(k)=sigma(k)).

See also A376187, A376188, A376190 for single lines.

Terms a(19) onward from Max Alekseyev, Aug 18 2024

A376188 For a line L in the plane, let C(L) denote the number of prime points [k, prime(k)] on L, and let M(L) denote the maximum prime(k) of any of these points; a(n) = minimum M(L) over all lines with C(L) >= n.

Original entry on oeis.org

2, 3, 7, 23, 47, 73, 73, 73, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 509, 4021, 4021, 4021, 4021, 4021, 4021, 4021, 4027, 4027, 4027, 4027, 26759, 26759, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947, 26947

Offset: 1

N. J. A. Sloane, Sep 23 2024

nonn

To avoid any confusion, C(L) is the total number of prime points on L, by definition.

See A376187 (which considers lines L with C(L) equal to n) for further information.

Max Alekseyev, Table of n, a(n) for n = 1..79

Cf. A005115, A373813, A376187, A376190.

a(21)-a(52) from Max Alekseyev, Sep 28 2024

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).

Original entry on oeis.org

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.

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.

A376488 a(n) is the least k such that A375422(k) = n.

Original entry on oeis.org

1, 2, 4, 9, 13, 16, 18, 21, 71, 72, 75, 77, 79, 82, 84, 85, 88, 93, 95, 97, 470, 472, 475, 497, 500, 511, 515, 526, 529, 544, 557, 2618, 2738, 2743, 2744, 2749, 2761, 2762, 2827, 2832, 2835, 2862, 2890, 2892, 2895, 2896, 2901, 2902, 2910, 2932, 2938, 2955

Offset: 1

Rémy Sigrist, Sep 25 2024

nonn

In other words, a(n) is the least k such that the set {(1, prime(1)), (2, prime(2)), ..., (k, prime(k))} contains n aligned points (where prime(k) denotes the k-th prime number).

Is this sequence infinite?

Rémy Sigrist, Table of n, a(n) for n = 1..102
Rémy Sigrist, PARI program

Cf. A375422, A376187, A376188.

PARI
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A373813 a(n) is the smallest number of straight lines needed to intersect all points (k, prime(k)) for k = 1..n.

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A376188 For a line L in the plane, let C(L) denote the number of prime points [k, prime(k)] on L, and let M(L) denote the maximum prime(k) of any of these points; a(n) = minimum M(L) over all lines with C(L) >= n.

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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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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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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).

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A376488 a(n) is the least k such that A375422(k) = n.

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