A373813 -id:A373813 - cp's OEIS frontend

A373814 RUNS transform of A373813.

2, 3, 4, 7, 8, 2, 5, 3, 5, 4, 3, 5, 8, 40, 28, 2, 8, 2, 2, 2, 7, 5, 3, 5, 4, 4, 9, 3, 14, 10, 5, 5, 3, 2, 3, 3, 3, 7, 5, 2, 5, 2, 10, 2, 4, 5, 3, 6, 6, 9, 4, 15, 5, 4, 2, 16, 5, 5, 5, 2, 11, 12, 10, 6, 5, 5

Offset: 1

N. J. A. Sloane, Aug 18 2024

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. A373812, A373813, A373815, A375515.

a(12)-a(66) from Max Alekseyev, Aug 20 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

A376187 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) = minimum M(L) over all lines with C(L) = n, or -1 if there is no such line.

Original entry on oeis.org

2, 3, 7, 23, 47, 181, 83, 73, 1069, 521, 701, 1627, 691, 4271, 4261, 3733, 3943, 3929, 10369, 509, 10463, 24683, 10259, 4297, 4159, 34963, 4021, 157907, 24923, 24691, 4027, 162007, 26759, 27283, 164821, 164503, 187721, 164839, 27067, 180437, 27143, 27059, 164663, 27043, 189961

Offset: 1

N. J. A. Sloane, Sep 23 2024

C(L) is the total number of prime points on L, by definition.
This sequence minimizes the largest prime in any line containing n prime-points. For the maximal smallest prime in any line that has the minimal largest prime (i.e. the lines arising in the present sequence), see A376190.
If C(L) = n in the definition is changed to C(L) >= n we get A376188.
Other known values are a(47) = 189887, a(48) = 164707, a(50)-a(58) = [180511, 180463, 26947, 193373, 180289, 180541, 164627, 194083, 186311], a(60) = 193871, a(62)-a(65) = [187471, 194239, 194309, 194141], a(67)-a(70) = [194269, 193723, 193513, 192737], a(76)-a(79) = [194069, 194267, 193789, 193841]. - Max Alekseyev, Sep 27 2024.

			The following are lines corresponding to a(1) to a(8). We describe the lines by simply listing the primes "prime(k)" corresponding to the points on the line.
  n   L
  1   2
  2   2,3
  3   3,5,7
  4   5,11,17,23
  5   19,23,31,43,47
  6   61,71,101,131,151,181
  7   7,11,59,67,71,79,83
  8   13,17,29,37,41,53,61,73
There are two parallel lines of slope 6 which both contain 20 points. The first contains the points with [x,y] coordinates [45, 197], [51, 233], [52, 239], [54, 251], [55, 257], [56, 263], [57, 269], [64, 311], [71, 353], [72, 359], [76, 383], [77, 389], [79, 401], [86, 443], [87, 449], [89, 461], [92, 479], [94, 491], [96, 503], [97, 509] (here y == -1 mod 6),
and the second contains the points [42, 181], [44, 193], [47, 211], [50, 229], [63, 307], [67, 331], [68,337], [70, 349], [73, 367], [74, 373], [75, 379], [78, 397], [80, 409],[82, 421], [84, 433], [85, 439], [88, 457], [93, 487], [95, 499], [99, 523] (here y == 1 mod 6).
The existence of these two lines was confirmed by _W. Edwin Clark_, who produced the illustration in the LINKS section. This shows an enlargement of the region 35 <= x <= 105. The blue dots are the points on the first line, the red dots those on the second line.
It is interesting to contrast these two 20-point lines with the results in A005115, which gives the earliest arithmetic progressions of primes with a given number of terms. To find an arithmetic progression of 20 primes one has to go out to 572945039351. Of course these primes don't lie on a line, because of the irregular spacing between the primes.
For many further examples of lines containing many prime-points see the Table in the LINKS section.
There are at least five lines of 54 points each and slope 12; and at least 56 lines of 18 points each and slope 12.  There is a 79-point line, connecting (12125,129533)-(17484,193841), again with slope 12. Populous slope-12 lines are common within my search range. - _Don Reble_, Oct 02 2024.

W. Edwin Clark, A line of slope 6 containing 20 prime-points (blue dots), and a parallel line, also with 20 prime-points (red dots)
N. J. A. Sloane, Table of lines in the plane containing the known maximum numbers of prime-points
N. J. A. Sloane, Sketch taken from A373813 which includes lines corresponding to a(3) = 7 and a(4) = 23
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. A005115, A373813, A376188, A376190.

a(9) corrected by Rémy Sigrist, Sep 24 2024.
a(12) from W. Edwin Clark, Sep 25 2024.
a(14)-a(45) from Max Alekseyev, Sep 26 2024, and independently confirmed by Don Reble, Oct 02 2024.

A373810 a(n) is the smallest number of straight lines needed to intersect all points (k, phi(k)) for k = 1..n (where phi is the Euler totient function A000010).

Original entry on oeis.org

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, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17

Offset: 1

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

nonn

Max Alekseyev, Table of n, a(n) for n = 1..455
Max Alekseyev, Sage program for lines covering points, Github, Aug 19 2024
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. A000010, A375499, A373813, A373815 (RUNS).

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

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

A375502 a(n) is the minimal number of parallel straight lines needed to intersect all points (k, prime(k)) for k = 1..n (where prime(k) denotes the k-th prime number).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 9, 9, 10, 11, 11, 12, 12, 13, 14, 14, 15, 16, 16, 16, 17, 18, 18, 19, 19, 19, 19, 20, 21, 21, 22, 22, 22, 22, 22, 22, 23, 24, 25, 25, 26, 26, 26, 26, 26, 26, 26, 26

Offset: 1

Rémy Sigrist, Aug 18 2024

nonn

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Rémy Sigrist, PARI program

Cf. A373811, A375433, A373813.

PARI
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\\ See Links section.
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A378831 The minimum number of diagonal lines required to cover all vertices created when the n outer vertices of a regular n-gon are connected by diagonal lines.

Original entry on oeis.org

1, 2, 2, 4, 5, 10, 11, 21

Offset: 2

Scott R. Shannon, Dec 08 2024

See A007569 for the total number of vertices created by the diagonal lines of a regular n-gon.
The terms a(9)..a(13) are less than or equal to 21, 25, 38, 41, 59 respectively.
Note that these numbers are not always the same as the minimum number of required lines if the lines can be arbitrarily placed. The first known occurrence where the numbers differ is for a(7) = 10 where one can use eight diagonal lines plus one off-diagonal line to cover two vertices in such a way that all vertices can be covered with only 9 lines.
Better upper bounds for a(10)..a(13) are 23, 36, 38, 56, with a(10), a(11) being almost certainly the correct values. - Scott R. Shannon, Dec 10 2024

Scott R. Shannon, Image showing one example of 11 diagonals covering all 57 vertices of the octagon. The covering diagonals are shown as thicker white lines.

Cf. A007569, A373813, A373811.

a(9) added by Scott R. Shannon, Dec 10 2024

cp's OEIS Frontend

A373814 RUNS transform of A373813.

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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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A376187 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) = minimum M(L) over all lines with C(L) = n, or -1 if there is no such line.

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A373810 a(n) is the smallest number of straight lines needed to intersect all points (k, phi(k)) for k = 1..n (where phi is the Euler totient function A000010).

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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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PARI

Extensions

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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A375502 a(n) is the minimal number of parallel straight lines needed to intersect all points (k, prime(k)) for k = 1..n (where prime(k) denotes the k-th prime number).

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PARI

A378831 The minimum number of diagonal lines required to cover all vertices created when the n outer vertices of a regular n-gon are connected by diagonal lines.

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