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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified March 4 08:38 EST 2021. Contains 341781 sequences. (Running on oeis4.)