A326333 - OEIS

%I #7 Jun 28 2019 21:14:23

%S 1,1,2,3,5,7,11,15,22,30,42,56,76,99,132,171,222,283,363,457,577,721,

%T 902,1115,1379,1693,2076,2530,3077,3723,4500,5410,6494,7765,9270,

%U 11025,13089,15491,18307,21569,25369,29765,34869,40750,47546,55361,64367,74685,86529

%N Number of integer partitions of n with sortable prime factors.

%C An integer partition has sortable prime factors if there is a permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

%F A000041(n) = a(n) + A326332(n).

%t Table[Length[Select[IntegerPartitions[n],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,0,20}]

%Y Unsortable integer partitions are A326332.

%Y Sortable normal multiset partitions are A326212.

%Y Sortable factorizations are A326334.

%Y Cf. A000041, A000108, A001055, A058681, A112798, A326209, A326211, A326258.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jun 27 2019