A376667 -id:A376667 - cp's OEIS frontend

A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 appears twice in row 8 of A036038 (and never more than twice in the same row): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!). Hence, a(56) = 2.

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

Cf. A036038, A376369, A376661, A376664, A376665 (records), A376666 (indices of records), A376667.

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.

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

A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

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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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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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