%I #6 Aug 01 2018 09:23:14

%S 10,14,15,20,21,22,24,26,28,33,34,35,38,39,40,42,44,45,46,48,50,51,52,

%T 54,55,56,57,58,62,63,65,66,68,69,70,72,74,75,76,77,78,80,82,84,85,86,

%U 87,88,91,92,93,94,95,96,98,99,100,102,104,105,106,108,110

%N Heinz numbers of integer partitions that are not uniformly normal.

%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.

%e Sequence of all non-uniformly normal integer partitions begins: (31), (41), (32), (311), (42), (51), (2111), (61), (411), (52), (71), (43), (81), (62), (3111), (421), (511), (322), (91), (21111), (331).

%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, A317493, A317588, A317589.

%K nonn

%O 1,1

%A _Gus Wiseman_, Aug 01 2018