1,1

Table of n, a(n) for n=1..64.

Cf. A090252, A354159, A354160, A354161, A354162.

nonn,new

N. J. A. Sloane, May 30 2022

approved

1,1

Table of n, a(n) for n=1..59.

Cf. A090252, A354159, A354160, A354161, A354163.

nonn,new

N. J. A. Sloane, May 30 2022

approved

1,1

Table of n, a(n) for n=1..64.

Cf. A090252, A354159, A354160, A354162, A354163.

nonn,new

N. J. A. Sloane, May 30 2022

approved

1,1

Table of n, a(n) for n=1..62.

Cf. A090252, A354159, A354161, A354162, A354163.

nonn,new

N. J. A. Sloane, May 30 2022

approved

2,8

The integer coordinates of the 4 vertices of the quadrilateral are (x1,0), (w,y2), (x3,h), (0,y4), 0 < x1, x3 < w, 0 < y2, y4 < h, such that the 6 distances between the 4 vertices are distinct.

The relationship to A353532 is that the number of lattice points n X m is used there, while here the side lengths of the lattice rectangle w = n - 1 and h = m - 1 are used.

Table of n, a(n) for n=2..79.

Hugo Pfoertner, PARI program to print sequence terms.

The triangle begins, with corresponding terms of A353532 shown in parenthesis:

   \ d 2       3       4        5        6        7        8       9

  w \---------------------------------------------------------------------

  2 |  0 ( 0)  |       |        |        |        |        |       |

  3 |  0 ( 0)  0 ( 0)  |        |        |        |        |       |

  4 |  0 ( 0)  1 ( 3)  0 (  1)  |        |        |        |       |

  5 |  1 ( 1)  3 ( 7)  1 ( 12)  0 ( 11)  |        |        |       |

  6 |  0 ( 1)  3 (11)  3 ( 26)  4 ( 52)  4 ( 40)  |        |       |

  7 |  3 ( 4)  6 (23)  6 ( 50)  6 ( 94) 12 (147)  0 (105)  |       |

  8 |  2 ( 4)  6 (30)  7 ( 69)  9 (127) 15 (198) 13 (301)  6 (190) |

  9 |  6 (10) 10 (49) 12 (103) 12 (192) 30 (302) 18 (444) 27 (583) 8 (379)

.

Only 1 = T(4,3) of the 3 = T_a353532(5,4) quadrilaterals has diagonals AC, BD whose intersection point S has integer coordinates:

.

   3 | . C . . .     3 | . C . . .     3 | . . C . .

   2 | . . . . .     2 | . . . . B     2 | . . . . B

   1 | D S . . B     1 | D . . . .     1 | D . . . .

   0 | . A . . .     0 | . A . . .     0 | . A . . .

   y /----------     y /----------     y /----------

     x 0 1 2 3 4       x 0 1 2 3 4       x 0 1 2 3 4

        S=(1,1)          S=(1,5/4)     S=(16/11,15/11)

.

T(5,2) = T_a353532(6,3) = 1:

.

   2 | . . . C . .

   1 | D . S . . B

   0 | . A . . . .

   y /------------

     x 0 1 2 3 4 5

        S=(2,1)

.

T(5,3) = 3 of the T_a353532(6,4) = 7 intersection points S of the diagonals AC, BD have integer coordinates:

.

  3 | . C . . . .   3 | . C . . . .   3 | . . C . . .   3 | . . . C . .

  2 | . . . . . .   2 | . . . . . B   2 | . . . . . .   2 | D . . . . .

  1 | D S . . . B   1 | D . . . . .   1 | D . . . . B   1 | . . . . . B

  0 | . A . . . .   0 | . A . . . .   0 | . A . . . .   0 | . A . . . .

  y /------------   y /------------   y /------------   y /------------

    x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5

       S=(1,1)           S=(1,6/5)         S=(4/3,1)     S=(35/17,27/17)

.

  3 | . . . . C .   3 | . . C . . .   3 | . . C . . .

  2 | . . . . . .   2 | . . . . . .   2 | . . . . . B

  1 | D . S . . B   1 | D . S . . B   1 | D . . . . .

  0 | . A . . . .   0 | . . A . . .   0 | . . A . . .

  y /------------   y /------------   y /------------

    x 0 1 2 3 4 5     x 0 1 2 3 4 5     x 0 1 2 3 4 5

       S=(2,1)           S=(2,1)           S=(2,7/5)

(PARI) see link. The program a354490 (w1, w2) prints the terms for the rows w1 .. w2. An auxiliary function sinter is defined to determine the rational intersection point of the diagonals.

Cf. A353532, A353533, A354488.

A354491 is the diagonal of the triangle.

nonn,tabl,new

Hugo Pfoertner, May 30 2022

approved

0,3

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

Table of n, a(n) for n=0..60.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The a(1) = 1 through a(8) = 8 partitions:

  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

       (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)

                    (211)            (222)                (422)

                    (1111)           (3111)               (2222)

                                     (111111)             (4211)

                                                          (41111)

                                                          (221111)

                                                          (11111111)

For example, the partition (3,2,2,2,1,1,1) has trajectory: (1,1,1,2,2,2,3) -> (3,3,6) -> (6,6) -> (12), so is counted under a(12).

Table[Length[Select[IntegerPartitions[n], Length[NestWhile[Sort[Total/@Split[#]]&, #, !UnsameQ@@#&]]<=1&]], {n, 0, 30}]

Dominated by A018818 (partitions into divisors).

The version for compositions is A353858.

A275870 counts collapsible partitions, ranked by A300273.

A304442 counts partitions with all equal run-sums, ranked by A353833.

A325268 counts partitions by omicron, rank statistic A304465.

A353832 represents the operation of taking run-sums of a partition.

A353837 counts partitions with all distinct run-sums, ranked by A353838.

A353840-A353846 pertain to partition run-sum trajectory.

A353847-A353859 pertain to composition run-sum trajectory.

A353864 counts rucksack partitions, ranked by A353866.

Cf. A000041, A008284, A181819, A225485, A325239, A325277, A325280, A326370, A353834, A353839, A353865.

nonn,new

Gus Wiseman, May 26 2022

approved

0,8

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking run-sums (or condensations) until a strict partition is reached. For example, the trajectory of (2,1,1) is (2,1,1) -> (2,2) -> (4).

Also the number of integer partitions of n with Kimberling's depth statistic (see A237685, A237750) equal to k-1.

Table of n, a(n) for n=0..77.

Triangle begins:

   1

   0   1

   0   1   1

   0   2   1   0

   0   2   2   1   0

   0   3   4   0   0   0

   0   4   6   1   0   0   0

   0   5   9   1   0   0   0   0

   0   6  11   4   1   0   0   0   0

   0   8  20   2   0   0   0   0   0   0

   0  10  25   7   0   0   0   0   0   0   0

   0  12  37   6   1   0   0   0   0   0   0   0

   0  15  47  13   2   0   0   0   0   0   0   0   0

   0  18  67  15   1   0   0   0   0   0   0   0   0   0

   0  22  85  25   3   0   0   0   0   0   0   0   0   0   0

   0  27 122  26   1   0   0   0   0   0   0   0   0   0   0   0

For example, row n = 8 counts the following partitions (empty columns indicated by dots):

.  (8)    (44)        (422)     (4211)  .  .  .  .

   (53)   (332)       (32111)

   (62)   (611)       (41111)

   (71)   (2222)      (221111)

   (431)  (3221)

   (521)  (3311)

          (5111)

          (22211)

          (311111)

          (2111111)

          (11111111)

rsn[y_]:=If[y=={}, {}, NestWhileList[Reverse[Sort[Total/@ Split[Sort[#]]]]&, y, !UnsameQ@@#&]];

Table[Length[Select[IntegerPartitions[n], Length[rsn[#]]==k&]], {n, 0, 15}, {k, 0, n}]

Row-sums are A000041.

Column k = 1 is A000009.

Column k = 2 is A237685.

Column k = 3 is A237750.

The version for run-lengths instead of run-sums is A225485 or A325280.

This statistic (trajectory length) is ranked by A353841 and A326371.

The version for compositions is A353859, see also A353847-A353858.

A005811 counts runs in binary expansion.

A275870 counts collapsible partitions, ranked by A300273.

A304442 counts partitions with all equal run-sums, ranked by A353833.

A353832 represents the operation of taking run-sums of a partition

A353836 counts partitions by number of distinct run-sums.

A353838 ranks partitions with all distinct run-sums, counted by A353837.

A353840-A353846 pertain to partition run-sum trajectory.

A353845 counts partitions whose run-sum trajectory ends in a singleton.

Cf. A008284, A047966, A181819, A325239, A325277, A353834, A353865.

nonn,tabl,new

Gus Wiseman, May 26 2022

approved

3,4

Table of n, a(n) for n=3..80.

Triangle begins:

    1;

    1,  1;

    2,  1,  1;

    2,  2,  1,  1;

    4,  3,  2,  1,  1;

    5,  4,  3,  2,  1,  1;

    9,  6,  5,  3,  2,  1, 1;

   13, 10,  6,  5,  3,  2, 1, 1;

   23, 15, 10,  7,  5,  3, 2, 1, 1;

   35, 24, 14, 10,  7,  5, 3, 2, 1, 1;

   61, 39, 23, 14, 11,  7, 5, 3, 2, 1, 1;

   98, 63, 34, 21, 14, 11, 7, 5, 3, 2, 1, 1;

In particular, the rooted trees counted in the first three rows of the triangle are shown by using the Hasse diagram as follows:

  ---------

    o   o

     \ /

      o

  ----------------------

    o       |

    |       |

    o   o   |   o  o  o

     \ /    |    \ | /

      o     |      o

  ------------------------------------------------------

    o   o        o   o   |   o         |

     \ /         |   |   |   |         |

      o   o      o   o   |   o  o  o   |   o  o   o  o

       \ /        \ /    |    \ | /    |    \  \ /  /

        o          o     |      o      |        o

Cf. A000081, A000598, A001190, A292556, A299038.

nonn,tabl,new

Salah Uddin Mohammad, Mar 17 2022

approved

1,2

a(1) = 1 since P(1) = 1 by convention. Without this convention we would have a(1) = 2.

a(5) <= 437489361912143559513287483711091603378 (De Koninck, 2009).

Table of n, a(n) for n=1..4.

Jean-Marie De Koninck, Those Fascinating Numbers, American Mathematical Society, 2009, p. 173, entry 6859.

Jean-Marie De Koninck and Matthieu Moineau, Consecutive Integers Divisible by a Power of their Largest Prime Factor, J. Integer Seq., Vol. 21 (2018), Article 18.9.3.

a(2) = 8 since 8 = 2^3, P(8) = 2 and 2^2|8, 9 = 3^2, P(9) = 3 and 3^2 | 9, and 8 is the least number with this property.

a(3) = 6859 since 6859 = 19^3, P(6859) = 19 and 19^3 | 6859, 6860 = 2^2 * 5 * 7^3, P(6860) = 7 and 7^3 | 6860, and 6859 is the least number with this property.

Cf. A006530, A071178, A354558, A354562.

nonn,more,bref,new

Amiram Eldar, May 30 2022

approved

1,1

De Koninck and Moineau (2018) proved that this sequence is infinite assuming the Bunyakovsky conjecture.

Jean-Marie De Koninck and Nicolas Doyon, The Life of Primes in 37 Episodes, American Mathematical Society, 2021, p. 232.

Table of n, a(n) for n=1..23.

Jean-Marie De Koninck and Matthieu Moineau, Consecutive Integers Divisible by a Power of their Largest Prime Factor, J. Integer Seq., Vol. 21 (2018), Article 18.9.3.

Eric Weisstein's World of Mathematics, Bouniakowsky Conjecture.

Wikipedia, Bunyakovsky conjecture.

101250 = 2 * 3^4 * 5^4 is a term since P(101250) = 5 and 5^4 | 101250, 101251 = 19 * 73^2, P(101251) = 73, and 73^2 | 101251.

p[n_] := FactorInteger[n][[-1, 2]]; Select[Range[3*10^7], p[#] > 3 && p[# + 1] > 1 &]

(Python)

from sympy import factorint

def c(n, e): f = factorint(n); return f[max(f)] >= e

def ok(n): return n > 1 and c(n, 4) and c(n+1, 2)

print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, May 30 2022

Subsequence of A070003, A354558 and A354564.

Cf. A006530, A071178, A354565.

nonn,new

Amiram Eldar, May 30 2022

approved