A376371 -id:A376371 - cp's OEIS frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

2, 3, 10, 4, 12, 6, 5, 15, 20, 420, 7, 21, 30, 630, 120, 8, 24, 56, 840, 1680, 210, 9, 28, 60, 1980, 60060, 1260, 4324320, 11, 35, 90, 3003, 83160, 2520, 21621600, 7207200, 13, 36, 105, 7140, 180180, 5040, 24504480, 151351200, 720720

Offset: 1

Pontus von Brömssen, Sep 22 2024

Row n lists numbers m such that A376369(m) = n.
In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
Any integer k >= 2 appears exactly once in the array.

			Array begins:
  n\k|       1         2         3         4         5          6          7          8
  ---+---------------------------------------------------------------------------------
  1  |       2         3         4         5         7          8          9         11
  2  |      10        12        15        21        24         28         35         36
  3  |       6        20        30        56        60         90        105        252
  4  |     420       630       840      1980      3003       7140       7560       9240
  5  |     120      1680     60060     83160    180180     240240     831600     900900
  6  |     210      1260      2520      5040     27720     166320    1441440    4084080
  7  | 4324320  21621600  24504480  43243200  75675600  116396280  367567200  908107200
  8  | 7207200 151351200 302702400 411863760 823727520 1816214400 2327925600 4655851200

Pontus von Brömssen, Table of n, a(n) for n = 1..1081 (antidiagonals 1..46)

Cf. A036038, A050382, A078760, A318762, A325472 (complement of first row), A325593 (complement of the union of the first 2 rows), A376369, A376376 (first column).

First five rows are A376371, A376372, A376373, A376374, A376375.

A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n.

Original entry on oeis.org

0, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 14, 6, 16, 17, 18, 19, 5, 7, 22, 23, 4, 25, 26, 27, 8, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 7, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 5, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Index of first row of A078760 (or A036038 when n >= 2) that contains n.

a(n) <= n, with equality if and only if n is in A376371, i.e., if and only if n is not in A325472.

			a(6) = 3, because 6 appears in row 3 of A078760, corresponding to the multinomial coefficient 3!/(1!*1!*1!) = 6.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Log-log plot, using Plot2.

Cf. A036038, A078760, A325472, A376369, A376370, A376371.

a(k!) = k for k != 1.

A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

Is this the same as A357759? - R. J. Mathar, Oct 09 2024. [Answer: No, they are different. - Andrew Howroyd, Oct 09 2024]

			56 is not a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8: 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

First row of A376667.
Complement of A325306 (with respect to the positive integers).
Cf. A036038, A078760, A376371, A376663.

cp's OEIS Frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

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A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n.

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A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

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