A334969 - OEIS

%I #8 Jun 09 2020 22:13:57

%S 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 31,32,33,34,35,36,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,

%U 61,62,64,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83

%N Heinz numbers of alternately strong integer partitions.

%C First differs from A304678 in lacking 450.

%C First differs from A316529 (the totally strong version) in having 150.

%C 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 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 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 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t altstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],altstrQ[Reverse[Length/@Split[q]]]]];

%t Select[Range[100],altstrQ[Reverse[primeMS[#]]]&]

%Y The co-strong version is A317257.

%Y The case of reversed partitions is (also) A317257.

%Y The total version is A316529.

%Y These partitions are counted by A332339.

%Y Totally co-strong partitions are counted by A332275.

%Y Alternately co-strong compositions are counted by A332338.

%Y Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 09 2020