A327410 Numbers represented by the partition coefficients of prime partitions.

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

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