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A316529 Heinz numbers of totally strong integer partitions.

%I A316529 #48 Apr 18 2023 05:25:14
%S A316529 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 A316529 31,32,33,34,35,36,37,38,39,41,42,43,46,47,49,50,51,53,54,55,57,58,59,
%U A316529 61,62,64,65,66,67,69,70,71,73,74,75,77,78,79,81,82,83
%N A316529 Heinz numbers of totally strong integer partitions.
%C A316529 First differs from A304678 at a(115) = 151, A304678(115) = 150.
%C A316529 The alternating version first differs from this sequence in having 150 and lacking 450.
%C A316529 An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
%C A316529 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 A316529 Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.
%e A316529 Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.
%t A316529 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A316529 totstrQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totstrQ[Length/@Split[q]]]];
%t A316529 Select[Range[100],totstrQ[Reverse[primeMS[#]]]&]
%Y A316529 Cf. A056239, A181819, A182850, A242031, A296150, A305732, A317246.
%Y A316529 The enumeration of these partitions by sum is A316496.
%Y A316529 The complement is A316597.
%Y A316529 The widely normal version is A332291.
%Y A316529 The dual version is A335376.
%Y A316529 Partitions with weakly decreasing run-lengths are A100882.
%Y A316529 Cf. A025487, A100883, A133808, A317257, A332275, A332293, A332297.
%K A316529 nonn
%O A316529 1,2
%A A316529 _Gus Wiseman_, Jul 29 2018
%E A316529 Updated with corrected terminology by _Gus Wiseman_, Mar 08 2020