A317493 -id:A317493 - cp's OEIS frontend

A317491 Number of fully normal integer partitions of n.

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 33, 46, 50, 68, 77, 100, 112, 146, 167, 201, 234, 290, 326, 400, 456, 545, 622, 744, 845, 1004, 1153, 1351, 1551, 1819, 2103, 2434, 2808, 3248, 3735, 4304, 4943, 5661, 6506, 7446, 8499, 9657, 11070, 12505, 14325, 16183

Offset: 0

Gus Wiseman, Jul 30 2018

nonn

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.

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

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

fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],fulnrmQ]],{n,0,30}]

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

A317492 Heinz numbers of fully normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

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.

Cf. A055932, A056239, A181819, A182850, A296150, A305732, A317089, A317090, A317245, A317246, A317491, A317493.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],fulnrmQ[Reverse[primeMS[#]]]&]

A317590 Heinz numbers of integer partitions that are not uniformly normal.

Original entry on oeis.org

10, 14, 15, 20, 21, 22, 24, 26, 28, 33, 34, 35, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 108, 110

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

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

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.

			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).

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317493, A317588, A317589.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];
Select[Range[1000],!uninrmQ[primeMS[#]]&]

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A317491 Number of fully normal integer partitions of n.

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A317492 Heinz numbers of fully normal integer partitions.

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A317590 Heinz numbers of integer partitions that are not uniformly normal.

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