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%I #5 Nov 14 2018 08:09:09
%S 2,6,8,30,36,40,48,64,210,252,270,280,300,324,336,360,400,432,448,480,
%T 576,640,768,1024,2310,2772,2940,2970,3080,3150,3300,3528,3564,3696,
%U 3780,3920,3960,4050,4200,4400,4500,4536,4704,4752,4860,4928,5040,5280,5400
%N Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%C These partitions are those that are finer than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.
%e The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (21), (111), (321), (2211), (3111), (21111), (111111), (4321), (42211), (32221), (43111), (33211), (222211), (421111), (322111), (331111), (2221111), (4111111), (3211111), (22111111), (31111111), (211111111), (1111111111).
%e The partition (322111) has Heinz number 360 and can be partitioned as ((1)(2)(3)(112)), ((1)(2)(12)(13)), or ((1)(11)(3)(22)), so 360 belongs to the sequence.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Select[Range[2,1000],Select[Map[Total[primeMS[#]]&,facs[#],{2}],Sort[#]==Range[Max@@#]&]!={}&]
%Y Subsequence of A242422.
%Y Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
%Y Cf. A321467, A321468, A321470, A321472, A321514.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 13 2018
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