cp's OEIS Frontend

A332293 Heinz numbers of widely totally co-strongly normal integer partitions.

%I A332293 #5 Feb 18 2020 04:47:44
%S A332293 1,2,4,6,8,12,16,30,32,64,128,180,210,256,360,512,1024,2048,2310,4096,
%T A332293 8192,16384,30030,32768,65536,75600,131072,262144,510510,524288
%N A332293 Heinz numbers of widely totally co-strongly normal integer partitions.
%C A332293 An integer partition is widely totally 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 are themselves a widely totally co-strongly normal partition.
%C A332293 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A332293 The sequence of terms together with their prime indices begins:
%e A332293      1: {}
%e A332293      2: {1}
%e A332293      4: {1,1}
%e A332293      6: {1,2}
%e A332293      8: {1,1,1}
%e A332293     12: {1,1,2}
%e A332293     16: {1,1,1,1}
%e A332293     30: {1,2,3}
%e A332293     32: {1,1,1,1,1}
%e A332293     64: {1,1,1,1,1,1}
%e A332293    128: {1,1,1,1,1,1,1}
%e A332293    180: {1,1,2,2,3}
%e A332293    210: {1,2,3,4}
%e A332293    256: {1,1,1,1,1,1,1,1}
%e A332293    360: {1,1,1,2,2,3}
%e A332293    512: {1,1,1,1,1,1,1,1,1}
%e A332293   1024: {1,1,1,1,1,1,1,1,1,1}
%e A332293   2048: {1,1,1,1,1,1,1,1,1,1,1}
%e A332293   2310: {1,2,3,4,5}
%e A332293   4096: {1,1,1,1,1,1,1,1,1,1,1,1}
%e A332293   8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
%e A332293 For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.
%t A332293 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332293 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A332293 gnaQ[y_]:=Or[y=={},Union[y]=={1},And[normQ[y],LessEqual@@Length/@Split[y],gnaQ[Length/@Split[y]]]];
%t A332293 Select[Range[1000],gnaQ[Reverse[primeMS[#]]]&]
%Y A332293 A subset of A055932.
%Y A332293 Closed under A181819.
%Y A332293 The non-co-strong version is A332276.
%Y A332293 The enumeration of these partitions by sum is A332278.
%Y A332293 The alternating version is A332290.
%Y A332293 The strong version is A332291.
%Y A332293 The case of reversed partitions is (also) A332291.
%Y A332293 Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.
%K A332293 nonn,more
%O A332293 1,2
%A A332293 _Gus Wiseman_, Feb 16 2020