%I #5 Jul 30 2018 22:16:00

%S 1,1,2,3,4,6,6,10,12,17,21,30,33,46,50,68,77,100,112,146,167,201,234,

%T 290,326,400,456,545,622,744,845,1004,1153,1351,1551,1819,2103,2434,

%U 2808,3248,3735,4304,4943,5661,6506,7446,8499,9657,11070,12505,14325,16183

%N Number of fully normal integer partitions of n.

%C An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

%F a(n) = A317245(n) iff n is 1 or a prime number.

%e The a(6) = 6 fully normal partitions are (6), (51), (42), (411), (321), (111111). Missing from this list are (33), (3111), (222), (2211), (21111).

%t fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];

%t Table[Length[Select[IntegerPartitions[n],fulnrmQ]],{n,0,30}]

%Y Cf. A181819, A182850, A182857, A275870, A304660, A305563, A317081, A317086, A317088, A317246, A317492, A317493.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jul 30 2018