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A305188
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Numbers that are equal to a nontrivial multinomial coefficient (i.e., equal to k!/(k1!*...*km!) with k1 + ... + km = k, k-2 >= k1 >= ... >= km).
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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)
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)
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(Python) # see link above
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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