A332277 -id:A332277 - cp's OEIS frontend

A332293 Heinz numbers of widely totally co-strongly normal integer partitions.

1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 16 2020

An integer partition is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal partition.

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

			The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    12: {1,1,2}
    16: {1,1,1,1}
    30: {1,2,3}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   256: {1,1,1,1,1,1,1,1}
   360: {1,1,1,2,2,3}
   512: {1,1,1,1,1,1,1,1,1}
  1024: {1,1,1,1,1,1,1,1,1,1}
  2048: {1,1,1,1,1,1,1,1,1,1,1}
  2310: {1,2,3,4,5}
  4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.

A subset of A055932.
Closed under A181819.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332278.
The alternating version is A332290.
The strong version is A332291.
The case of reversed partitions is (also) A332291.
Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
gnaQ[y_]:=Or[y=={},Union[y]=={1},And[normQ[y],LessEqual@@Length/@Split[y],gnaQ[Length/@Split[y]]]];
Select[Range[1000],gnaQ[Reverse[primeMS[#]]]&]

A332272 Number of narrowly recursively normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 14, 18, 23, 30, 37, 46, 52, 70, 80, 100, 116, 146, 171, 203, 236, 290, 332, 401, 458, 547, 626, 744, 851, 1004, 1157, 1353, 1553, 1821, 2110, 2434, 2810, 3250, 3741, 4304, 4949, 5661, 6510, 7450, 8501, 9657, 11078, 12506, 14329, 16185

Offset: 0

Gus Wiseman, Mar 08 2020

nonn

A sequence is narrowly recursively normal if either it is constant (narrow) or its run-lengths are a narrowly recursively normal sequence covering an initial interval of positive integers (normal).

			The a(6) = 8 partitions are (6), (51), (42), (411), (33), (321), (222), (111111). Missing from this list are (3111), (2211), (21111).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (111111)  (511)      (431)
                                               (3211)     (521)
                                               (1111111)  (611)
                                                          (2222)
                                                          (3221)
                                                          (4211)
                                                          (11111111)

The strict instead of narrow version is A330937.
The normal case is A332277.
The widely normal case is A332277(n) - 1 for n > 1.
The wide version is A332295(n) - 1.
Cf. A000009, A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332576.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[Length[qtn]<=1,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

For n > 1, a(n) = A317491(n) + A000005(n) - 2.

A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 31, 35, 51, 59, 80, 97, 130, 153, 204, 244, 308, 376, 475, 564, 708, 851, 1043, 1247, 1533, 1816, 2216, 2633, 3174, 3766, 4526, 5324, 6376, 7520, 8917, 10479, 12415, 14524, 17134, 20035, 23489, 27423, 32091, 37286, 43512

Offset: 0

Gus Wiseman, Mar 05 2020

nonn

First differs from A317491 at a(11) = 31, A317491(11) = 30.

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (43)       (53)
             (111)  (211)   (41)     (51)      (52)       (62)
                    (1111)  (221)    (321)     (61)       (71)
                            (311)    (411)     (322)      (332)
                            (11111)  (111111)  (331)      (422)
                                               (421)      (431)
                                               (511)      (521)
                                               (3211)     (611)
                                               (1111111)  (3221)
                                                          (4211)
                                                          (11111111)

The narrow version is A317081.
Heinz numbers of these partitions first differ from A317492 in having 420.
Not counting constant-1 sequences gives A317081.
Dominated by A332295.
Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332272, A332277.

nQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},Union[Length/@Split[ptn]]==Range[Max[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n],nQ]],{n,0,30}]

a(n > 1) = A317081(n) + 1.

A330937 Number of strictly recursively normal integer partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 20, 27, 35, 49, 58, 81, 100, 126, 160, 206, 246, 316, 374, 462, 564, 696, 813, 1006, 1195, 1441, 1701, 2058, 2394, 2896, 3367, 4007, 4670, 5542, 6368, 7540, 8702, 10199, 11734, 13760, 15734, 18384, 21008, 24441, 27893, 32380, 36841

Offset: 0

Gus Wiseman, Mar 09 2020

nonn

A sequence is strictly recursively normal if either it empty, its run-lengths are distinct (strict), or its run-lengths cover an initial interval of positive integers (normal) and are themselves a strictly recursively normal sequence.

			The a(1) = 1 through a(9) = 15 partitions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
            (21)  (31)   (32)   (42)   (43)    (53)    (54)
                  (211)  (41)   (51)   (52)    (62)    (63)
                         (221)  (321)  (61)    (71)    (72)
                         (311)  (411)  (322)   (332)   (81)
                                       (331)   (422)   (432)
                                       (421)   (431)   (441)
                                       (511)   (521)   (522)
                                       (3211)  (611)   (531)
                                               (3221)  (621)
                                               (4211)  (711)
                                                       (3321)
                                                       (4221)
                                                       (4311)
                                                       (5211)
                                                       (32211)

The narrow instead of strict version is A332272.
A wide instead of strict version is A332295(n) - 1 for n > 1.
Cf. A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332277, A332576.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[ptn=={},UnsameQ@@qtn,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

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A332293 Heinz numbers of widely totally co-strongly normal integer partitions.

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A332272 Number of narrowly recursively normal integer partitions of n.

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A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

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A330937 Number of strictly recursively normal integer partitions of n.

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