cp's OEIS Frontend

A327410 Numbers represented by the partition coefficients of prime partitions.

%I A327410 #16 Feb 16 2025 08:33:58
%S A327410 1,6,10,20,21,36,56,78,90,105,120,171,210,252,300,364,465,528,560,741,
%T A327410 756,792,903,990,1140,1176,1485,1540,1680,1830,1953,1980,2346,2520,
%U A327410 2600,2628,2775,3240,3432,3570,4095,4368,4851,4960,5253,5460,5886,5984,6105
%N A327410 Numbers represented by the partition coefficients of prime partitions.
%C A327410 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 A327410 A partition is a prime partition if all parts are prime.
%H A327410 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 A327410 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%e A327410 (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 A327410 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 A327410 1   <- [2],
%e A327410 6   <- [2, 2],
%e A327410 10  <- [3, 2],
%e A327410 20  <- [3, 3],
%e A327410 21  <- [5, 2],
%e A327410 36  <- [7, 2],
%e A327410 56  <- [5, 3],
%e A327410 78  <- [11, 2],
%e A327410 90  <- [2, 2, 2],
%e A327410 105 <- [13, 2],
%e A327410 120 <- [7, 3],
%e A327410 171 <- [17, 2],
%e A327410 210 <- [3, 2, 2],
%e A327410 252 <- [5, 5],
%e A327410 300 <- [23, 2].
%o A327410 (SageMath)
%o A327410 def A327410_list(n):
%o A327410     res = []
%o A327410     for k in range(2*n):
%o A327410         P = Partitions(k, parts_in = prime_range(k+1))
%o A327410         res += [multinomial(p) for p in P]
%o A327410     return sorted(Set(res))[:n]
%o A327410 print(A327410_list(20))
%Y A327410 Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.
%K A327410 nonn
%O A327410 1,2
%A A327410 _Peter Luschny_, Sep 07 2019