A381432 -id:A381432 - cp's OEIS frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

0, 1, 2, 2, 4, 5, 7, 10, 13, 16, 21, 28, 33, 45, 55, 65, 83, 105, 121, 155, 180, 217, 259, 318, 362, 445, 512, 614, 707, 850, 958, 1155, 1309, 1543, 1754, 2079, 2327, 2740, 3085, 3592, 4042, 4699, 5253, 6093, 6815, 7839, 8751, 10069, 11208, 12832, 14266, 16270

Offset: 0

Clark Kimberling and Peter J. C. Moses, Mar 19 2014

nonn

Suppose that p = x(1) >= x(2) >= ... >= x(k) is a partition of n. Let y(1) > y(2) > ... > y(h) be the distinct parts of p, and let m(i) be the multiplicity of y(i) for 1 <= i <= h. Then we can "look" at p as "m(1) y(1)'s and m(2) y(2)'s and ... m(h) y(h)'s". Reversing the m's and y's, we can then "say" the Look-and-Say partition of p, denoted by LS(p). The name "Look-and-Say" follows the example of Look-and-Say integer sequences (e.g., A005150). As p ranges through the partitions of n, LS(p) ranges through all the Look-and-Say partitions of n. The number of these is A239455(n).
The Look-and-Say array is distinct from the Wilf array, described at A098859; for example, the number of Look-and-Say partitions of 9 is A239455(9) = 16, whereas the number of Wilf partitions of 9 is A098859(9) = 15. The Look-and-Say partition of 9 which is not a Wilf partition of 9 is [2,2,2,1,1,1].
Conjecture: a partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are all distinct, so y is counted under a(9). - Gus Wiseman, Aug 11 2025
Also the number of integer partitions y of n such that there is a pairwise disjoint way to choose a strict integer partition of each multiplicity (or run-length) of y. - Gus Wiseman, Aug 11 2025

			The 11 partitions of 6 generate 7 Look-and-Say partitions as follows:
6 -> 111111
51 -> 111111
42 -> 111111
411 -> 21111
33 -> 222
321 -> 111111
3111 -> 3111
222 -> 33
2211 -> 222
21111 -> 411
111111 -> 6,
so that a(6) counts these 7 partitions: 111111, 21111, 222, 3111, 33, 411, 6.

These include all Wilf partitions, counted by A098859, ranked by A130091.
These partitions are listed by A239454 in graded reverse-lex order.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351204, complement A351203.
The complement is counted by A351293, apparently ranked by A351295, conjugate A381433.
These partitions appear to be ranked by A351294, conjugate A381432.
The non-Wilf case is counted by A351592.
For normal multisets we appear to have A386580, complement A386581.
A000110 counts set partitions, ordered A000670.
A000569 = graphical partitions, complement A339617.
A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
A181819 = Heinz number of the prime signature of n (prime shadow).
A279790 counts disjoint families on strongly normal multisets.
A329738 = compositions with all equal run-lengths.
A386583 counts separable partitions, sums A325534, ranks A335433.
A386584 counts inseparable partitions, sums A325535, ranks A335448.
A386585 counts separable type partitions, sums A336106, ranks A335127.
A386586 counts inseparable type partitions, sums A386638 or A025065, ranks A335126.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018, ranked by A044813.
- A329739 = compositions, for runs A351013, ranked by A351596.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A297770, A304660, A305563, A329746, A351201, A351202, A351291.

LS[part_List] := Reverse[Sort[Flatten[Map[Table[#[[2]], {#[[1]]}] &, Tally[part]]]]]; LS[n_Integer] := #[[Reverse[Ordering[PadRight[#]]]]] &[DeleteDuplicates[Map[LS, IntegerPartitions[n]]]]; TableForm[t = Map[LS[#] &, Range[10]]](*A239454,array*)
Flatten[t](*A239454,sequence*)
Map[Length[LS[#]] &, Range[25]](*A239455*)
(* Peter J. C. Moses, Mar 18 2014 *)
disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]>0&]],{n,0,10}] (* Gus Wiseman, Aug 11 2025 *)

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],!MemberQ[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]&]

A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 3, 0, 4, 1, 1, 0, 5, 0, 6, 0, 2, 2, 8, 0, 2, 2, 0, 0, 10, 1, 12, 0, 2, 3, 2, 0, 15, 3, 2, 0, 18, 1, 22, 0, 0, 5, 27, 0, 2, 0, 3, 0, 32, 0, 3, 0, 4, 5, 38, 0, 46, 7, 0, 0, 4, 1, 54, 0, 5, 1, 64, 0, 76, 8, 0, 0, 3, 1, 89, 0, 0, 10

Offset: 1

Gus Wiseman, May 15 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 25 are (3,3), for which we have choices ((3),(2,1)) and ((2,1),(3)), so a(25) = 2.
The prime indices of 91 are (4,6), for which we have choices ((4),(6)), ((4),(5,1)), ((4),(3,2,1)), ((3,1),(6)), ((3,1),(4,2)), so a(91) = 5.
The prime indices of 273 are (2,4,6), for which we have choices ((2),(4),(6)), ((2),(4),(5,1)), ((2),(3,1),(6)), so a(273) = 3.

Adding up over all integer partitions gives A279790, strict A279375.
Without disjointness we have A357982, non-strict version A299200.
For multiplicities instead of indices we have A382525.
Positions of 0 appear to be A382912, counted by A383710, odd case A383711.
Positions of positive terms are A382913, counted by A383708, odd case A383533.
Positions of 1 are A383707, counted by A179009.
The conjugate version is A384005.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A098859, A122111, A130091, A209229, A317141, A381454, A382771, A382876.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
prix[n_]:=If[n==1,{}, Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[pof[prix[n]]],{n,100}]

A381431 Heinz number of the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 10, 13, 11, 11, 16, 17, 15, 19, 14, 13, 13, 23, 20, 25, 17, 27, 22, 29, 13, 31, 32, 17, 19, 17, 25, 37, 23, 19, 28, 41, 17, 43, 26, 33, 29, 47, 40, 49, 35, 23, 34, 53, 45, 19, 44, 29, 31, 59, 26, 61, 37, 39, 64, 23, 19, 67, 38

Offset: 1

Gus Wiseman, Feb 26 2025

nonn

The image first differs from A320340, A364347, A350838 in containing a(150) = 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
   7: {4}
  11: {5}
  10: {1,3}
  13: {6}
  11: {5}
  11: {5}
  16: {1,1,1,1}

The conjugate is A048767, union A351294, complement A351295, fix A048768 (count A217605).
Taking length instead of sum in the definition gives A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
The union is A381432, complement A381433.
Values appearing only once are A381434, more than once A381435.
These are the Heinz numbers of rows of A381436, conjugate A381440.
Greatest prime index of each term is A381437, counted by A381438.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A005117, A047966, A051903, A066328, A091602, A116861, A130091, A212166, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Times@@Prime/@egs[prix[n]],{n,100}]

A122111(a(n)) = A048767(n).

A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 9, 11, 16, 20, 30, 34, 50, 58, 79, 96, 129, 152, 203, 243, 307, 375, 474, 563, 707, 850, 1042, 1246, 1532, 1815, 2215, 2632, 3173, 3765, 4525, 5323, 6375, 7519, 8916, 10478, 12414, 14523, 17133, 20034, 23488, 27422, 32090, 37285, 43511, 50559

Offset: 0

Gus Wiseman, Jul 21 2018

nonn

Also the number of integer partitions of n with distinct section-sums, where the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. - Gus Wiseman, Apr 21 2025

			The a(1) = 1 through a(9) = 16 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)

Chai Wah Wu, Table of n, a(n) for n = 0..160

The case with parts also covering an initial interval is A317088.
These partitions are ranked by A317090.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047966 counts partitions with constant section-sums.
A048767 interchanges prime indices and prime multiplicities (Look-and-Say), see A048768.
A055932 lists numbers whose prime indices cover an initial interval.
A116540 counts normal set multipartitions.
A304442 counts partitions with equal run-sums, ranks A353833.
A381436 lists the section-sum partition of prime indices.
A381440 lists the Look-and-Say partition of prime indices.
Cf. A000837, A069799, A217605, A317082, A317084, A317085, A317087.
Cf. A003242, A116861, A239455, A353837, A364916, A381431, A381432, A381438.

normalQ[m_]:=Union[m]==Range[Max[m]];
Table[Length[Select[IntegerPartitions[n],normalQ[Length/@Split[#]]&]],{n,30}]

from sympy.utilities.iterables import partitions
def A317081(n):
    if n == 0:
        return 1
    c = 0
    for d in partitions(n):
        s = set(d.values())
        if len(s) == max(s):
            c += 1
    return c # Chai Wah Wu, Jun 22 2020

A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Apr 05 2025

nonn

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.
Also the number of ways to choose a set of disjoint strict integer partitions, one of each nonzero multiplicity in the prime factorization of n.

			The a(27) = 2 partitions with Look-and-Say partition (2,2,2) are: (3,3), (2,2,1,1).
The prime indices of 3456 are {1,1,1,1,1,1,1,2,2,2}, and the partitions with Look-and-Say partition (2,2,2,1,1,1,1,1,1,1) are:
  (7,3,3)
  (7,2,2,1,1)
  (6,3,3,1)
  (5,3,3,2)
  (4,3,3,2,1)
  (4,3,2,2,1,1)
so a(3456) = 6.

Positions of positive terms are A351294, conjugate A381432.
Positions of 0 are A351295, conjugate A381433.
Positions of 1 are A381540, conjugate A381434.
Positions of terms > 1 are A381541, conjugate A381435.
Positions of first appearances are A382775.
A000670 counts ordered set partitions.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 lists the section-sum partition of prime indices, ranks A381431.
A381440 lists the Look-and-Say partition of prime indices, ranks A048767.
Cf. A003557, A047966, A048768, A050361, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964.

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
Table[Length[stp[Last/@FactorInteger[n]]],{n,100}]

a(2^n) = A000009(n).

a(prime(n)) = 1.

A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 5, 7, 8, 13, 14, 18, 22, 27, 36, 41, 50, 61, 73, 86

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions y of n whose normal multiset (in which i appears y_i times) is a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(1) = 1 through a(9) = 8 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (4,2,1)  (7,1)    (8,1)
                                                   (4,3,1)  (4,3,2)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)

These partitions have Heinz numbers A382913.
Without ones we have A383533, complement A383711.
The number of such families for each Heinz number is A383706.
The complement is counted by A383710, ranks A382912.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A091602, A130091, A317141, A351013, A381441, A382771, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], pof[#]!={}&]],{n,15}]

A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 1, 1, 3, 4, 6, 10, 15, 22, 29, 42, 59, 79, 108, 140, 190, 247, 324, 417, 541

Offset: 0

Gus Wiseman, May 07 2025

Also the number of integer partitions of n whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (221)    (222)     (322)      (332)
               (211)   (311)    (411)     (331)      (422)
               (1111)  (2111)   (2211)    (511)      (611)
                       (11111)  (3111)    (2221)     (2222)
                                (21111)   (3211)     (3221)
                                (111111)  (4111)     (3311)
                                          (22111)    (4211)
                                          (31111)    (5111)
                                          (211111)   (22211)
                                          (1111111)  (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

These partitions have Heinz numbers A382912.
The number of such families for each Heinz number is A383706.
The complement is counted by A383708, ranks A382913.
Without ones we have A383711, complement A383533.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], pof[#]=={}&]], {n,0,15}]

A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

Original entry on oeis.org

4, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32, 36, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135, 136, 140, 144, 148, 150, 152, 153, 156, 160, 162, 164

Offset: 1

Gus Wiseman, Apr 12 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms, prime indices, and corresponding multisets begin:
   4:       {1,1} {1,2}
   8:     {1,1,1} {1,2,3}
   9:       {2,2} {1,1,2,2}
  12:     {1,1,2} {1,1,2,3}
  16:   {1,1,1,1} {1,2,3,4}
  18:     {1,2,2} {1,1,2,2,3}
  20:     {1,1,3} {1,1,1,2,3}
  24:   {1,1,1,2} {1,1,2,3,4}
  27:     {2,2,2} {1,1,2,2,3,3}
  28:     {1,1,4} {1,1,1,1,2,3}
  32: {1,1,1,1,1} {1,2,3,4,5}
  36:   {1,1,2,2} {1,1,2,2,3,4}
  40:   {1,1,1,3} {1,1,1,2,3,4}
  44:     {1,1,5} {1,1,1,1,1,2,3}
  45:     {2,2,3} {1,1,1,2,2,3,3}
  48: {1,1,1,1,2} {1,1,2,3,4,5}
  50:     {1,3,3} {1,1,1,2,2,2,3}
  52:     {1,1,6} {1,1,1,1,1,1,2,3}

The Look-and-Say partition is ranked by A048767, listed by A381440.
Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293.
For prime indices instead of signature we have A351295, conjugate A381433.
The complement is A382913.
For equal instead of distinct run-lengths we have A382914, see A382858, A382879, A382915.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A381431 lists the section-sum partition of n, ranks A381436, union A381432.
Cf. A000720, A001221, A001222, A181821, A305936, A335125, A351202, A382525, A382771, A382773, A382775, A382857.
Cf. A381636, A381717, A381871, A382876.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{}, Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],Not@*lasQ@*nrmptn]

A382913 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has a permutation with all distinct run-lengths.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103

Offset: 1

Gus Wiseman, Apr 12 2025

nonn

This described multiset (row n of A305936, Heinz number A181821) is generally not the same as the multiset of prime indices of n (A112798). For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239.

			The terms, prime indices, and corresponding multisets begin:
   1:    {} {}
   2:   {1} {1}
   3:   {2} {1,1}
   5:   {3} {1,1,1}
   6: {1,2} {1,1,2}
   7:   {4} {1,1,1,1}
  10: {1,3} {1,1,1,2}
  11:   {5} {1,1,1,1,1}
  13:   {6} {1,1,1,1,1,1}
  14: {1,4} {1,1,1,1,2}
  15: {2,3} {1,1,1,2,2}
  17:   {7} {1,1,1,1,1,1,1}
  19:   {8} {1,1,1,1,1,1,1,1}
  21: {2,4} {1,1,1,1,2,2}
  22: {1,5} {1,1,1,1,1,2}
  23:   {9} {1,1,1,1,1,1,1,1,1}
  25: {3,3} {1,1,1,2,2,2}
  26: {1,6} {1,1,1,1,1,1,2}

Look-and-Say partitions are counted by A239455, ranks A351294.
Non-Look-and-Say partitions are counted by A351293, ranks A351295.
For prime indices instead of signature we have A351294, conjugate A381432.
The Look-and-Say partition of n is listed by A381440, rank A048767.
The complement is A382912.
For equal run-lengths we have the complement of A382914, see A382858, A382879, A382915.
A044813 lists numbers whose binary expansion has distinct run-lengths.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A381431 ranks section-sum partition, listed by A381436.
Cf. A000720, A001221, A001222, A140690, A181821, A305936, A335125, A382525, A382771, A382773, A382775.
Cf. A381636, A381717, A381871, A382876.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&, If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_} :> Table[PrimePi[p],{k}]]]]];
lasQ[y_]:=Select[Permutations[y], UnsameQ@@Length/@Split[#]&]!={};
Select[Range[100],lasQ@*nrmptn]

cp's OEIS Frontend

A239455 Number of Look-and-Say partitions of n; see Comments.

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A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

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A383706 Number of ways to choose disjoint strict integer partitions, one of each prime index of n.

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A381431 Heinz number of the section-sum partition of the prime indices of n.

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A317081 Number of integer partitions of n whose multiplicities cover an initial interval of positive integers.

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A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.

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A383708 Number of integer partitions of n such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A383710 Number of integer partitions of n such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A382912 Numbers k such that row k of A305936 (a multiset whose multiplicities are the prime indices of k) has no permutation with all distinct run-lengths.

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