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A335377 Heinz numbers of non-totally co-strong integer partitions.

%I A335377 #6 Jun 05 2020 09:57:26
%S A335377 18,50,54,60,75,84,90,98,108,120,126,132,140,147,150,156,162,168,198,
%T A335377 204,220,228,234,240,242,245,250,260,264,270,276,280,294,300,306,308,
%U A335377 312,315,324,336,338,340,342,348,350,363,364,372,375,378,380,408,414,420
%N A335377 Heinz numbers of non-totally co-strong integer partitions.
%C A335377 A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.
%C A335377 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 A335377 The sequence of terms together with their prime indices begins:
%e A335377    18: {1,2,2}        156: {1,1,2,6}        276: {1,1,2,9}
%e A335377    50: {1,3,3}        162: {1,2,2,2,2}      280: {1,1,1,3,4}
%e A335377    54: {1,2,2,2}      168: {1,1,1,2,4}      294: {1,2,4,4}
%e A335377    60: {1,1,2,3}      198: {1,2,2,5}        300: {1,1,2,3,3}
%e A335377    75: {2,3,3}        204: {1,1,2,7}        306: {1,2,2,7}
%e A335377    84: {1,1,2,4}      220: {1,1,3,5}        308: {1,1,4,5}
%e A335377    90: {1,2,2,3}      228: {1,1,2,8}        312: {1,1,1,2,6}
%e A335377    98: {1,4,4}        234: {1,2,2,6}        315: {2,2,3,4}
%e A335377   108: {1,1,2,2,2}    240: {1,1,1,1,2,3}    324: {1,1,2,2,2,2}
%e A335377   120: {1,1,1,2,3}    242: {1,5,5}          336: {1,1,1,1,2,4}
%e A335377   126: {1,2,2,4}      245: {3,4,4}          338: {1,6,6}
%e A335377   132: {1,1,2,5}      250: {1,3,3,3}        340: {1,1,3,7}
%e A335377   140: {1,1,3,4}      260: {1,1,3,6}        342: {1,2,2,8}
%e A335377   147: {2,4,4}        264: {1,1,1,2,5}      348: {1,1,2,10}
%e A335377   150: {1,2,3,3}      270: {1,2,2,2,3}      350: {1,3,3,4}
%e A335377 For example, 60 is the Heinz number of (3,2,1,1), which has run-lengths: (1,1,2) -> (2,1) -> (1,1) -> (2) -> (1). Since (2,1) is not weakly increasing, 60 is in the sequence.
%t A335377 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A335377 totcostrQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totcostrQ[Length/@Split[q]]]];
%t A335377 Select[Range[100],!totcostrQ[Reverse[primeMS[#]]]&]
%Y A335377 Partitions with weakly increasing run-lengths are counted by A100883.
%Y A335377 Totally strong partitions are counted by A316496.
%Y A335377 Heinz numbers of totally strong partitions are A316529.
%Y A335377 The version for reversed partitions is A316597.
%Y A335377 The strong version is (also) A316597.
%Y A335377 The alternating version is A317258.
%Y A335377 Totally co-strong partitions are counted by A332275.
%Y A335377 The complement is A335376.
%Y A335377 Cf. A100882, A181819, A182850, A242031, A305732, A317256, A329747, A332291, A332293, A332339.
%K A335377 nonn
%O A335377 1,1
%A A335377 _Gus Wiseman_, Jun 05 2020