cp's OEIS Frontend

A317143 In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.

%I A317143 #7 Jul 22 2018 22:54:27
%S A317143 1,2,3,4,4,5,6,8,6,8,7,9,10,12,16,8,9,12,16,10,12,16,11,14,15,18,20,
%T A317143 24,32,12,16,13,21,22,25,27,28,30,36,40,48,64,14,18,20,24,32,15,18,20,
%U A317143 24,32,16,17,26,33,35,42,44,45,50,54,56,60,72,80,96,128
%N A317143 In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.
%C A317143 The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C A317143 If x and y are partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.
%e A317143 The partitions finer than or equal to (2,2) are (2,2), (2,1,1), (1,1,1,1), with Heinz numbers 9, 12, 16, so the 9th row is {9, 12, 16}.
%e A317143 Triangle begins:
%e A317143    1
%e A317143    2
%e A317143    3   4
%e A317143    4
%e A317143    5   6   8
%e A317143    6   8
%e A317143    7   9  10  12  16
%e A317143    8
%e A317143    9  12  16
%e A317143   10  12  16
%e A317143   11  14  15  18  20  24  32
%e A317143   12  16
%e A317143   13  21  22  25  27  28  30  36  40  48  64
%e A317143   14  18  20  24  32
%e A317143   15  18  20  24  32
%e A317143   16
%e A317143   17  26  33  35  42  44  45  50  54  56  60  72  80  96 128
%t A317143 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A317143 Table[Union[Times@@@Map[Prime,Join@@@Tuples[IntegerPartitions/@primeMS[n]],{2}]],{n,12}]
%Y A317143 Row lengths are A300383.
%Y A317143 Cf. A002846, A056239, A213427, A215366, A265947, A296150, A299201, A317141.
%K A317143 nonn,tabf
%O A317143 1,2
%A A317143 _Gus Wiseman_, Jul 22 2018