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A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

%I A332290 #8 Feb 16 2020 07:54:45
%S A332290 1,2,4,6,8,12,16,30,32,60,64,128,210,256,360,512,1024,2048,2310,2520,
%T A332290 4096,8192,16384,30030,32768,65536,75600,131072,262144,510510,524288
%N A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.
%C A332290 An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.
%C A332290 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A332290 This sequence is closed under A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.
%e A332290 The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
%e A332290       1: ()
%e A332290       2: (1)
%e A332290       4: (1,1)
%e A332290       6: (2,1)
%e A332290       8: (1,1,1)
%e A332290      12: (2,1,1)
%e A332290      16: (1,1,1,1)
%e A332290      30: (3,2,1)
%e A332290      32: (1,1,1,1,1)
%e A332290      60: (3,2,1,1)
%e A332290      64: (1,1,1,1,1,1)
%e A332290     128: (1,1,1,1,1,1,1)
%e A332290     210: (4,3,2,1)
%e A332290     256: (1,1,1,1,1,1,1,1)
%e A332290     360: (3,2,2,1,1,1)
%e A332290     512: (1,1,1,1,1,1,1,1,1)
%e A332290    1024: (1,1,1,1,1,1,1,1,1,1)
%e A332290    2048: (1,1,1,1,1,1,1,1,1,1,1)
%e A332290    2310: (5,4,3,2,1)
%e A332290    2520: (4,3,2,2,1,1,1)
%e A332290 For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.
%t A332290 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332290 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
%t A332290 Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]
%Y A332290 Closed under A181821.
%Y A332290 The non-co-strong version is A332276.
%Y A332290 The enumeration of these partitions by sum is A332289.
%Y A332290 The total (rather than alternating) version is A332293.
%Y A332290 Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.
%K A332290 nonn,more
%O A332290 1,2
%A A332290 _Gus Wiseman_, Feb 14 2020