

A327410


Numbers represented by the partition coefficients of prime partitions.


0



1, 6, 10, 20, 21, 36, 56, 78, 90, 105, 120, 171, 210, 252, 300, 364, 465, 528, 560, 741, 756, 792, 903, 990, 1140, 1176, 1485, 1540, 1680, 1830, 1953, 1980, 2346, 2520, 2600, 2628, 2775, 3240, 3432, 3570, 4095, 4368, 4851, 4960, 5253, 5460, 5886, 5984, 6105
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OFFSET

1,2


COMMENTS

Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by <pi>. We say 'k is represented by pi' if k = <pi>.
A partition is a prime partition if all parts are prime.


LINKS

Table of n, a(n) for n=1..49.
George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition


EXAMPLE

(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1 < [2],
6 < [2, 2],
10 < [3, 2],
20 < [3, 3],
21 < [5, 2],
36 < [7, 2],
56 < [5, 3],
78 < [11, 2],
90 < [2, 2, 2],
105 < [13, 2],
120 < [7, 3],
171 < [17, 2],
210 < [3, 2, 2],
252 < [5, 5],
300 < [23, 2].


PROG

(SageMath)
def A327410_list(n):
res = []
for k in range(2*n):
P = Partitions(k, parts_in = prime_range(k+1))
res += [multinomial(p) for p in P]
return sorted(Set(res))[:n]
print(A327410_list(20))


CROSSREFS

Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.
Sequence in context: A257858 A095985 A270306 * A145351 A227874 A015783
Adjacent sequences: A327407 A327408 A327409 * A327411 A327412 A327413


KEYWORD

nonn


AUTHOR

Peter Luschny, Sep 07 2019


STATUS

approved



