A373810 -id:A373810 - cp's OEIS frontend

A373815 RUNS transform of A373810.

2, 3, 3, 5, 4, 6, 4, 7, 5, 9, 6, 8, 5, 16, 7, 8, 6, 16, 5, 10, 11, 7, 15, 6, 12, 8, 9, 17, 21, 10, 14, 7, 13, 13, 39, 7, 12, 6, 11, 17, 8, 17, 13, 26

Offset: 1

N. J. A. Sloane, Aug 18 2024

Cf. A373810, A373812, A373814, A375515.

a(11)-a(44) from Max Alekseyev, Aug 20 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

A373811 a(0) = 0. For n > 0, a(n) is the smallest number of straight lines needed to intersect all points (k, a(k)) for 0 <= k < n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18

Offset: 0

N. J. A. Sloane, Aug 13 2024, based on an email from Dominic McCarty

The github site of Arthur O'Dwyer has illustrations of many of the small configurations of lines. At his suggestion, I am including his drawings for n = 5, 8, 13, 17, 23, 28, which are just before a(n) increases.

Dominic McCarty, Email to N. J. A. Sloane, Aug 13 2024.

Max Alekseyev, Table of n, a(n) for n = 0..400
Max Alekseyev, Sage program for lines covering points, Github, Aug 19 2024
Dominic McCarty, Equations entailing a(0) to a(31)
Daniel Mondot, Upper bounds on a(n) for n < 10000
Arthur O'Dwyer, A373811, Github. Includes Python program and many illustrations.
Arthur O'Dwyer, Illustration for a(5)
Arthur O'Dwyer, Illustration for a(8)
Arthur O'Dwyer, Illustration for a(13)
Arthur O'Dwyer, Illustration for a(17)
Arthur O'Dwyer, Illustration for a(23)
Arthur O'Dwyer, Illustration for a(28)
N. J. A. Sloane, Sketch illustrating a(8) = 3
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]

See A373812 for the lengths of runs of identical terms.

For minimal sets of lines covering some classic sequences, see A373810, A373813, A375499.

a(32)-a(46) from Zachary DeStefano, Aug 14 2024

a(35) corrected and terms a(47) onward added by Max Alekseyev, Aug 15 2024

A375499 a(n) is the smallest number of straight lines needed to intersect all points (k, d(k)) for k = 1..n (where d is the sum-of-divisors function A000005).

Original entry on oeis.org

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

Offset: 1

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

nonn

			The initial terms, together with an appropriate set of lines, are:
  1 1 [1]
  2 1 [x]
  3 2 [2, x]
  4 2 [2, (2/3)*x + 1/3]
  5 2 [2, (2/3)*x + 1/3]
  6 3 [2, 2*x - 8, (2/3)*x + 1/3]
  7 3 [2, 2*x - 8, (2/3)*x + 1/3]
  8 3 [2, 4, (2/3)*x + 1/3]
  9 4 [2, 3, 4, x]
  10 4 [2, 3, 4, x]
  11 4 [2, 3, 4, x]
  12 4 [2, 3, 4, (5/11)*x + 6/11]
  13 4 [2, 3, 4, (5/11)*x + 6/11]
  14 4 [2, 3, 4, (5/11)*x + 6/11]
  15 4 [2, 3, 4, (5/11)*x + 6/11]
  16 5 [2, 3, 4, 4*x - 42, (4/15)*x + 11/15]
  17 5 [2, 3, 4, 4*x - 42, (4/15)*x + 11/15]
  18 5 [2, 3, 4, 6, (4/15)*x + 11/15]

Max Alekseyev, Table of n, a(n) for n = 1..400
Max Alekseyev, Sage program for lines covering points, Github, Aug 19 2024
Rémy Sigrist, PARI program
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]

Suggested by A373811 and A375420.

Cf. A000005, A373810, A373813, A375515 (RUNS).

PARI
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Terms a(30) onward from Max Alekseyev, Aug 18 2024

cp's OEIS Frontend

A373815 RUNS transform of A373810.

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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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A373811 a(0) = 0. For n > 0, a(n) is the smallest number of straight lines needed to intersect all points (k, a(k)) for 0 <= k < n.

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A375499 a(n) is the smallest number of straight lines needed to intersect all points (k, d(k)) for k = 1..n (where d is the sum-of-divisors function A000005).

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