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A334969 Heinz numbers of alternately strong integer partitions.

%I A334969 #8 Jun 09 2020 22:13:57
%S A334969 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,18,19,21,22,23,25,26,27,29,30,
%T A334969 31,32,33,34,35,36,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,
%U A334969 61,62,64,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83
%N A334969 Heinz numbers of alternately strong integer partitions.
%C A334969 First differs from A304678 in lacking 450.
%C A334969 First differs from A316529 (the totally strong version) in having 150.
%C A334969 A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately strong sequence.
%C A334969 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%e A334969 The sequence does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.
%t A334969 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A334969 altstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],altstrQ[Reverse[Length/@Split[q]]]]];
%t A334969 Select[Range[100],altstrQ[Reverse[primeMS[#]]]&]
%Y A334969 The co-strong version is A317257.
%Y A334969 The case of reversed partitions is (also) A317257.
%Y A334969 The total version is A316529.
%Y A334969 These partitions are counted by A332339.
%Y A334969 Totally co-strong partitions are counted by A332275.
%Y A334969 Alternately co-strong compositions are counted by A332338.
%Y A334969 Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.
%K A334969 nonn
%O A334969 1,2
%A A334969 _Gus Wiseman_, Jun 09 2020