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%I #15 Sep 09 2019 12:48:40
%S 1,6,10,20,21,36,56,78,90,105,120,171,210,252,300,364,465,528,560,741,
%T 756,792,903,990,1140,1176,1485,1540,1680,1830,1953,1980,2346,2520,
%U 2600,2628,2775,3240,3432,3570,4095,4368,4851,4960,5253,5460,5886,5984,6105
%N Numbers represented by the partition coefficients of prime partitions.
%C 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>.
%C A partition is a prime partition if all parts are prime.
%H George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, <a href="http://arxiv.org/abs/math/0509470">On the number of distinct multinomial coefficients</a>, arXiv:math/0509470 [math.CO], 2005.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%e (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.
%e 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.
%e 1 <- [2],
%e 6 <- [2, 2],
%e 10 <- [3, 2],
%e 20 <- [3, 3],
%e 21 <- [5, 2],
%e 36 <- [7, 2],
%e 56 <- [5, 3],
%e 78 <- [11, 2],
%e 90 <- [2, 2, 2],
%e 105 <- [13, 2],
%e 120 <- [7, 3],
%e 171 <- [17, 2],
%e 210 <- [3, 2, 2],
%e 252 <- [5, 5],
%e 300 <- [23, 2].
%o (SageMath)
%o def A327410_list(n):
%o res = []
%o for k in range(2*n):
%o P = Partitions(k, parts_in = prime_range(k+1))
%o res += [multinomial(p) for p in P]
%o return sorted(Set(res))[:n]
%o print(A327410_list(20))
%Y Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.
%K nonn
%O 1,2
%A _Peter Luschny_, Sep 07 2019
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