A376663 -id:A376663 - cp's OEIS frontend

A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

1, 2, 56, 3, 210, 166320, 4, 504, 360360, 4084080, 5, 1260, 720720, 17907120, 1396755360, 6, 1365, 2162160, 73513440, 4190266080, 698377680, 7, 1680, 5045040, 75675600, 4655851200, 13967553600, 146659312800, 8, 1716, 5765760, 220540320, 4942365120, 27935107200, 293318625600, 1075501627200

Offset: 1

Pontus von Brömssen, Oct 02 2024

In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

Each positive integer appears exactly once in the array, so as a linear sequence it is a permutation of the positive integers (unless there are any 0's).

			Array begins:
  n\k|             1             2              3              4              5              6
  ---+----------------------------------------------------------------------------------------
  1  |             1             2              3              4              5              6
  2  |            56           210            504           1260           1365           1680
  3  |        166320        360360         720720        2162160        5045040        5765760
  4  |       4084080      17907120       73513440       75675600      220540320      411863760
  5  |    1396755360    4190266080     4655851200     4942365120     9884730240    24443218800
  6  |     698377680   13967553600    27935107200   267711444000   537750813600   586637251200
  7  |  146659312800  293318625600  1606268664000  3226504881600  6184134356400  7228208988000
  8  | 1075501627200 6453009763200 12368268712800 24736537425600 29683844910720 74209612276800

Pontus von Brömssen, Table of n, a(n) for n = 1..465 (antidiagonals 1..30)

Cf. A036038, A078760, A325306 (complement of first row), A376370, A376663, A376673 (first column).

First five rows are A376668, A376669, A376670, A376671, A376672.

A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

Original entry on oeis.org

1, 56, 166320, 4084080, 1396755360, 698377680, 146659312800, 1075501627200, 37104806138400, 3710480613840000, 296838449107200, 86825246363856000, 96472495959840000, 36466603472819520000, 35251050023725536000, 272194921062320256000, 408292381593480384000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

After a(36), the sequence continues (where "?" represents terms that are either 0 or greater than 10^29): ?, 3059734941813910128088320000, ?, ?, 64254433778092112689854720000. After a(41), all terms are either 0 or greater than 10^29.

The terms a(1), a(3), ..., a(15), a(24), a(26), ..., a(36), a(38), a(41) are all in A025487, but a(16), ..., a(23), a(25) are all divisible by 17^2 but not by 13^2.

			First few terms and their representations as multinomial coefficients (corresponding to partitions with sum A376664(n)):
  a(1) =          1 = 0!;
  a(2) =         56 = 8!/(1!*1!*6!) = 8!/(3!*5!);
  a(3) =     166320 = 12!/(1!*1!*1!*4!*5!) = 12!/(1!*1!*2!*2!*6!) = 12!/(2!*2!*3!*5!);
  a(4) =    4084080 = 17!/(1!*1!*1!*4!*10!) = 17!/(1!*2!*5!*9!) = 17!/(2!*2!*3!*10!) = 17!/(4!*6!*7!);
  a(5) = 1396755360 = 19!/(1!*1!*1!*1!*1!*4!*10!) = 19!/(1!*1!*1!*2!*5!*9!) = 19!/(1!*1!*2!*2!*3!*10!) = 19!/(1!*1!*4!*6!*7!) = 19!/(3!*4!*5!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..36

First column of A376667.

Cf. A025487, A036038, A078760, A376376, A376663, A376664.

A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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, Oct 02 2024

nonn

Differs from A376368 at n = 1260, 2520, 5040, 60060, 83160, ... . For example, 1260 appears first in row A376368(1260) = 7 of A036038, but only once. It also appears once in row 9, but in row a(1260) = 10 it appears A376663(1260) = 2 times.

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

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

Cf. A036038, A325472, A376368, A376663.

A376666 Indices of records in A376663.

Original entry on oeis.org

1, 56, 166320, 4084080, 698377680, 146659312800, 1075501627200, 37104806138400, 296838449107200, 86825246363856000, 96472495959840000, 35251050023725536000, 272194921062320256000, 408292381593480384000, 4082923815934803840000, 15106818118958774208000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..25

Cf. A376378, A376663, A376665, A376673.

a(n) = A376673(A376665(n)).

A376665 Records in A376663.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 18, 20, 21, 23, 25, 28, 30, 31, 33, 38, 41

Offset: 1

Pontus von Brömssen, Oct 02 2024

Cf. A376377, A376663, A376666, A376673.

a(n) = A376663(A376666(n)).

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.

A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

Original entry on oeis.org

56, 210, 504, 1260, 1365, 1680, 1716, 2520, 5040, 7560, 9240, 13860, 15120, 17550, 21840, 24024, 25200, 25740, 27720, 30030, 42504, 43680, 55440, 60060, 69300, 72072, 75600, 77520, 83160, 110880, 120120, 151200, 154440, 168168, 180180, 185640, 203490, 240240

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 is a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8 (and never for more than 2 different partitions of the same integer): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

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

Second row of A376667.
Subsequence of A325306.
Cf. A036038, A078760, A376372, A376663.

A376670 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 3, i.e., numbers m such that A376663(m) = 3.

Original entry on oeis.org

166320, 360360, 720720, 2162160, 5045040, 5765760, 6683040, 7207200, 12252240, 14414400, 15135120, 24504480, 30270240, 35814240, 36756720, 37837800, 40360320, 40840800, 46558512, 49008960, 51482970, 61261200, 86486400, 98017920, 102965940, 110270160, 116396280

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			166320 is a term, because it can be represented as a multinomial coefficient for 3 different partitions of 12 (and never for more than 3 different partitions of the same integer): 166320 = 12!/(1!*1!*1!*4!*5!) = 12!/(1!*1!*2!*2!*6!) = 12!/(2!*2!*3!*5!).

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

Third row of A376667.

Cf. A036038, A078760, A376373, A376663.

A376671 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 4, i.e., numbers m such that A376663(m) = 4.

Original entry on oeis.org

4084080, 17907120, 73513440, 75675600, 220540320, 411863760, 1102701600, 1210809600, 2162049120, 2205403200, 2327925600, 2471182560, 3087564480, 5145940800, 6983776800, 8380532160, 9777287520, 10291881600, 10296594000, 19554575040, 20583763200, 20593188000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			4084080 is a term, because it can be represented as a multinomial coefficient for 4 different partitions of 17 (and never for more than 4 different partitions of the same integer): 4084080 = 17!/(1!*1!*1!*4!*10!) = 17!/(1!*2!*5!*9!) = 17!/(2!*2!*3!*10!) = 17!/(4!*6!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..9085 (all terms below 10^29)

Fourth row of A376667.

Cf. A036038, A078760, A376374, A376663.

A376672 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 5, i.e., numbers m such that A376663(m) = 5.

Original entry on oeis.org

1396755360, 4190266080, 4655851200, 4942365120, 9884730240, 24443218800, 48886437600, 61779564000, 83805321600, 97772875200, 107550162720, 123559128000, 247118256000, 412275623760, 419026608000, 535422888000, 803134332000, 879955876800, 899510451840, 1173274502400

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			1396755360 is a term, because it can be represented as a multinomial coefficient for 5 different partitions of 19 (and never for more than 5 different partitions of the same integer): 1396755360 = 19!/(1!*1!*1!*1!*1!*4!*10!) = 19!/(1!*1!*1!*2!*5!*9!) = 19!/(1!*1!*2!*2!*3!*10!) = 19!/(1!*1!*4!*6!*7!) = 19!/(3!*4!*5!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..5086 (all terms below 10^29)

Fifth row of A376667.

Cf. A036038, A078760, A376375, A376663.

cp's OEIS Frontend

A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

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A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

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A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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A376666 Indices of records in A376663.

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A376665 Records in A376663.

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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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A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

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A376670 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 3, i.e., numbers m such that A376663(m) = 3.

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A376671 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 4, i.e., numbers m such that A376663(m) = 4.

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A376672 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 5, i.e., numbers m such that A376663(m) = 5.

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