cp's OEIS Frontend

A332276 Heinz numbers of widely totally normal integer partitions.

%I A332276 #9 Feb 14 2020 08:02:47
%S A332276 1,2,4,6,8,12,16,18,30,32,60,64,90,128,150,180,210,256,300,360,450,
%T A332276 512,540,600,630,1024,1050,1350,1500,2048,2100,2250,2310,2520,2940,
%U A332276 3150,3780,4096,4200,4410,5880,8192,8820,9450,10500,11550,12600,13230,14700
%N A332276 Heinz numbers of widely totally normal integer partitions.
%C A332276 First differs from A317246 in having 630.
%C A332276 A sequence of positive integers is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
%C A332276 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e A332276 The sequence of terms together with their prime indices begins:
%e A332276     1: {}
%e A332276     2: {1}
%e A332276     4: {1,1}
%e A332276     6: {1,2}
%e A332276     8: {1,1,1}
%e A332276    12: {1,1,2}
%e A332276    16: {1,1,1,1}
%e A332276    18: {1,2,2}
%e A332276    30: {1,2,3}
%e A332276    32: {1,1,1,1,1}
%e A332276    60: {1,1,2,3}
%e A332276    64: {1,1,1,1,1,1}
%e A332276    90: {1,2,2,3}
%e A332276   128: {1,1,1,1,1,1,1}
%e A332276   150: {1,2,3,3}
%e A332276   180: {1,1,2,2,3}
%e A332276   210: {1,2,3,4}
%e A332276   256: {1,1,1,1,1,1,1,1}
%e A332276   300: {1,1,2,3,3}
%e A332276   360: {1,1,1,2,2,3}
%e A332276 For example, starting with (4,3,2,2,1), the partition with Heinz number 630, and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1). These are all normal and the last is all 1's, so 630 belongs to the sequence.
%t A332276 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A332276 gnaQ[y_]:=Or[y=={},Union[y]=={1},And[Union[y]==Range[Max[y]],gnaQ[Length/@Split[y]]]];
%t A332276 Select[Range[1000],gnaQ[primeMS[#]]&]
%Y A332276 Contains all powers of two A000079 and the primorials A002110.
%Y A332276 Heinz numbers of normal integer partitions are A055932.
%Y A332276 The case of reversed integer partitions is A332276 (this sequence).
%Y A332276 The enumeration of these partitions by sum is A332277.
%Y A332276 The enumeration of the generalization to compositions is A332279.
%Y A332276 The co-strong version is A332290.
%Y A332276 The strong version is A332291.
%Y A332276 Cf. A007097, A056239, A133808, A181819, A182850, A305732, A317081, A317089, A317090, A317246, A317492, A329747, A332295, A332296.
%K A332276 nonn
%O A332276 1,2
%A A332276 _Gus Wiseman_, Feb 12 2020