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A332291 Heinz numbers of widely totally strongly normal integer partitions.

%I A332291 #6 Feb 16 2020 07:54:52
%S A332291 1,2,4,6,8,16,18,30,32,64,128,210,256,450,512,1024,2048,2250,2310,
%T A332291 4096,8192,16384,30030,32768,65536,131072,262144,510510,524288
%N A332291 Heinz numbers of widely totally strongly normal integer partitions.
%C A332291 An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally strongly normal partition.
%C A332291 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%C A332291 This sequence is closed under A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.
%e A332291 The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
%e A332291       1: ()
%e A332291       2: (1)
%e A332291       4: (1,1)
%e A332291       6: (2,1)
%e A332291       8: (1,1,1)
%e A332291      16: (1,1,1,1)
%e A332291      18: (2,2,1)
%e A332291      30: (3,2,1)
%e A332291      32: (1,1,1,1,1)
%e A332291      64: (1,1,1,1,1,1)
%e A332291     128: (1,1,1,1,1,1,1)
%e A332291     210: (4,3,2,1)
%e A332291     256: (1,1,1,1,1,1,1,1)
%e A332291     450: (3,3,2,2,1)
%e A332291     512: (1,1,1,1,1,1,1,1,1)
%e A332291    1024: (1,1,1,1,1,1,1,1,1,1)
%e A332291    2048: (1,1,1,1,1,1,1,1,1,1,1)
%e A332291    2250: (3,3,3,2,2,1)
%e A332291    2310: (5,4,3,2,1)
%e A332291    4096: (1,1,1,1,1,1,1,1,1,1,1,1)
%t A332291 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332291 totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],GreaterEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
%t A332291 Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]
%Y A332291 Closed under A304660.
%Y A332291 The non-strong version is A332276.
%Y A332291 The co-strong version is A332293.
%Y A332291 The case of reversed partitions is (also) A332293.
%Y A332291 Heinz numbers of normal partitions with decreasing run-lengths are A025487.
%Y A332291 Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.
%K A332291 nonn,more
%O A332291 1,2
%A A332291 _Gus Wiseman_, Feb 14 2020