A317589 - OEIS

%I #6 Aug 01 2018 09:23:07

%S 1,2,3,4,5,6,7,8,9,11,12,13,16,17,18,19,23,25,27,29,30,31,32,36,37,41,

%T 43,47,49,53,59,60,61,64,67,71,73,79,81,83,89,90,97,101,103,107,109,

%U 113,121,125,127,128,131,137,139,149,150,151,157,163,167,169

%N Heinz numbers of uniformly normal integer partitions.

%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

%C An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];

%t Select[Range[1000],uninrmQ[primeMS[#]]&]

%Y Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317492, A317588, A317590.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 01 2018