A305188 Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!*...*km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).

6, 10, 12, 15, 20, 21, 24, 28, 30, 35, 36, 42, 45, 55, 56, 60, 66, 70, 72, 78, 84, 90, 91, 105, 110, 120, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 210, 220, 231, 240, 252, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 360, 364

Offset: 1

Author

Vincent Champain, May 27 2018

Keywords

• nonn

Comments

This sequence answers the following question: what numbers correspond to the number of permutations of a number of items that is lower than the number of permutations itself? It means that the underlying structure has some form of redundance / symmetry.

It can be shown that:

- no prime number is part of this sequence (see A304938)

- some nonprimes are not part of the sequence (beginning with 1, 4, 8, 9, 14, 16, 18, ...)

- any number that is a factorial of another integer e is part of this sequence (k=e, k1=ki=...=ke=1).

This sequence is a generalization of A006987.

From David A. Corneth, May 28 2018: (Start)

Also the numbers that are the number of permutations of either:

- sets of balls with two distinct colors of balls where each color occurs at least twice;

- sets of balls with at least three distinct colors of balls.

(End)

From Amiram Eldar, Jul 23 2020: (Start)

The asymptotic density of this sequence is 0 (Niven, 1951).

The number of terms not exceeding x is (1 + sqrt(2)) * x^(1/2) + o(x^(1/2)) (Erdős, 1954). (End)

Examples

			a(1) = 6 because all numbers lower than 6 are either prime or a power of primes.
105 is a term of the sequence because 105 is equal to a multinomial coefficient: 105 = (4+2+1)! / (4! * 2! * 1!) and 105 is the number of ways 7 balls can be sorted where 4 are red, 2 are yellow and one is blue.
2016 is a term because 64! / (62! * 2!) = 2016. - _David A. Corneth_, May 29 2018
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Vincent Champain, Calculation of the terms of the sequence including a k1..km list so that a(n) = k!/(k1!*...*km!) with k1+...+km = k and m > 1, 2018.
• Vincent Champain, Python program for A305188
• David A. Corneth, Terms with the corresponding tuples.
• Paul Erdős, The Number of Multinomial Coefficients, The American Mathematical Monthly, Vol. 61, No. 1 (1954), pp. 37-39.
• Ivan Niven, The asymptotic density of sequences, Bull. Amer. Math. Soc., Vol. 57 (1951), pp. 420-434. See theorem 2, p. 428.

Crossrefs

Cf. A008480, A304938, A006987.

Programs

Mathematica

mult[w_] := Total[w]!/Times @@ (w!); L = {}; Do[ t = mult /@ Select[ IntegerPartitions@ n, #[[1]] < n-1 &]; L = Union[L, Select[t, # <= 400 &]], {n, 3, 30}]; L (* Terms < 400, Giovanni Resta, May 27 2018 *)

Python

# see link above

Extensions

a(28)-a(57) from Giovanni Resta, May 27 2018