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%I #13 Jun 06 2020 10:47:03
%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,27,
%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,51,
%U 52,53,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70
%N Heinz numbers of alternately co-strong integer partitions.
%C The first term absent from this sequence but present in A242031 is 180.
%C A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e The sequence of terms together with their prime indices begins:
%e 1: {} 16: {1,1,1,1} 32: {1,1,1,1,1}
%e 2: {1} 17: {7} 33: {2,5}
%e 3: {2} 19: {8} 34: {1,7}
%e 4: {1,1} 20: {1,1,3} 35: {3,4}
%e 5: {3} 21: {2,4} 36: {1,1,2,2}
%e 6: {1,2} 22: {1,5} 37: {12}
%e 7: {4} 23: {9} 38: {1,8}
%e 8: {1,1,1} 24: {1,1,1,2} 39: {2,6}
%e 9: {2,2} 25: {3,3} 40: {1,1,1,3}
%e 10: {1,3} 26: {1,6} 41: {13}
%e 11: {5} 27: {2,2,2} 42: {1,2,4}
%e 12: {1,1,2} 28: {1,1,4} 43: {14}
%e 13: {6} 29: {10} 44: {1,1,5}
%e 14: {1,4} 30: {1,2,3} 45: {2,2,3}
%e 15: {2,3} 31: {11} 46: {1,9}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t totincQ[q_]:=Or[Length[q]<=1,And[OrderedQ[Length/@Split[q]],totincQ[Reverse[Length/@Split[q]]]]];
%t Select[Range[100],totincQ[Reverse[primeMS[#]]]&]
%Y Cf. A056239, A100883, A181819, A182850, A242031, A296150, A305732, A317246.
%Y These partitions are counted by A317256.
%Y The complement is A317258.
%Y Totally co-strong partitions are counted by A332275.
%Y Alternately co-strong compositions are counted by A332338.
%Y Alternately co-strong reversed partitions are counted by A332339.
%Y The total version is A335376.
%Y Cf. A182857, A304660, A305563, A316496, A332292, A332340.
%K nonn
%O 1,2
%A _Gus Wiseman_, Jul 25 2018
%E Updated with corrected terminology by _Gus Wiseman_, Jun 04 2020
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