A130091 -id:A130091 - cp's OEIS frontend

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 28, 44, 56, 80, 111, 148, 192, 264, 335, 447, 575, 743, 937, 1213, 1513, 1924, 2396, 3011, 3715, 4646, 5687, 7040, 8600, 10556, 12804, 15650, 18897, 22930, 27593, 33296, 39884, 47921, 57168, 68360, 81295, 96807, 114685

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).

See A239455 for the definition of Look-and-Say partitions.

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)  (42)    (43)    (53)     (54)
              (41)  (51)    (52)    (62)     (63)
                    (321)   (61)    (71)     (72)
                    (2211)  (421)   (431)    (81)
                            (3211)  (521)    (432)
                                    (3221)   (531)
                                    (3311)   (621)
                                    (4211)   (3321)
                                    (32111)  (4221)
                                             (4311)
                                             (5211)
                                             (32211)
                                             (42111)
                                             (321111)

The complement is counted by A239455, ranked by A351294.
These are all non-Wilf partitions (counted by A336866, ranked by A130092).
A variant for runs is A351203, complement A351204, ranked by A351201.
These partitions appear to be ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A000569 = graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n],Length[disjointFamilies[#]]==0&]],{n,0,15}] (* Gus Wiseman, Aug 13 2025 *)

a(n) = A000041(n) - A239455(n).

Edited by Gus Wiseman, Aug 12 2025

A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140

Offset: 1

Gus Wiseman, Feb 16 2022

nonn

First differs from A130092 (non-Wilf partitions) in lacking 216.

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 terms together with their prime indices begin:
      6: (2,1)         46: (9,1)         84: (4,2,1,1)
     10: (3,1)         51: (7,2)         85: (7,3)
     14: (4,1)         55: (5,3)         86: (14,1)
     15: (3,2)         57: (8,2)         87: (10,2)
     21: (4,2)         58: (10,1)        90: (3,2,2,1)
     22: (5,1)         60: (3,2,1,1)     91: (6,4)
     26: (6,1)         62: (11,1)        93: (11,2)
     30: (3,2,1)       65: (6,3)         94: (15,1)
     33: (5,2)         66: (5,2,1)       95: (8,3)
     34: (7,1)         69: (9,2)        100: (3,3,1,1)
     35: (4,3)         70: (4,3,1)      102: (7,2,1)
     36: (2,2,1,1)     74: (12,1)       105: (4,3,2)
     38: (8,1)         77: (5,4)        106: (16,1)
     39: (6,2)         78: (6,2,1)      110: (5,3,1)
     42: (4,2,1)       82: (13,1)       111: (12,2)
For example, the prime indices of 150 are {1,2,3,3}, with permutations and run-lengths (right):
  (3,3,2,1) -> (2,1,1)
  (3,3,1,2) -> (2,1,1)
  (3,2,3,1) -> (1,1,1,1)
  (3,2,1,3) -> (1,1,1,1)
  (3,1,3,2) -> (1,1,1,1)
  (3,1,2,3) -> (1,1,1,1)
  (2,3,3,1) -> (1,2,1)
  (2,3,1,3) -> (1,1,1,1)
  (2,1,3,3) -> (1,1,2)
  (1,3,3,2) -> (1,2,1)
  (1,3,2,3) -> (1,1,1,1)
  (1,2,3,3) -> (1,1,2)
Since none have all distinct run-lengths, 150 is in the sequence.

Wilf partitions are counted by A098859, ranked by A130091.
Non-Wilf partitions are counted by A336866, ranked by A130092.
A variant for runs is A351201, counted by A351203 (complement A351204).
These partitions appear to be counted by A351293.
The complement is A351294, apparently counted by A239455.
A032020 = number of binary expansions with distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A056239 = sum of prime indices, row sums of A112798.
A165413 = number of distinct run-lengths in binary expansion.
A181819 = Heinz number of prime signature (prime shadow).
A182850/A323014 = frequency depth, counted by A225485/A325280.
A297770 = number of distinct runs in binary expansion.
A320922 ranks graphical partitions, complement A339618, counted by A000569.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A329747 = runs-resistance, counted by A329746.
A333489 ranks anti-runs, complement A348612.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
Cf. A000961, A001221, A001222, A175413, A182857, A304660, A320924, A328592, A351202, A351290.

Select[Range[100],Select[Permutations[Join@@ ConstantArray@@@FactorInteger[#]],UnsameQ@@Length/@Split[#]&]=={}&]

Name edited by Gus Wiseman, Aug 13 2025

A381432 Heinz numbers of section-sum partitions. Union of A381431.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in 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:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
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, A047966, A051903, A066328, A116861, A130091, A212166, 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[#]]],#]&]

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[#]]],#]&]

A353837 Number of integer partitions of n with all distinct run-sums.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 14, 17, 28, 35, 49, 62, 85, 107, 149, 174, 238, 305, 384, 476, 614, 752, 950, 1148, 1451, 1763, 2205, 2654, 3259, 3966, 4807, 5773, 7039, 8404, 10129, 12140, 14528, 17288, 20668, 24505, 29062, 34437, 40704, 48059, 56748, 66577, 78228

Offset: 0

Gus Wiseman, May 26 2022

nonn

The run-sums of a sequence are the sums of its maximal consecutive constant subsequences (runs). For example, the run-sums of (2,2,1,1,1,3,2,2) are (4,3,3,4). The first partition whose run-sums are not all distinct is (2,1,1).

			The a(0) = 1 through a(6) = 10 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (1111)  (221)    (51)
                                (311)    (222)
                                (2111)   (321)
                                (11111)  (411)
                                         (2211)
                                         (21111)
                                         (111111)

Max Alekseyev, Table of n, a(n) for n = 0..100

For multiplicities instead of run-sums we have A098859, ranked by A130091.
For equal run-sums we have A304442, ranked by A353833 (nonprime A353834).
These partitions are ranked by A353838, complement A353839.
The version for compositions is A353850, ranked by A353852.
The weak version (rucksack partitions) is A353864, ranked by A353866.
The weak perfect version is A353865, ranked by A353867.
A005811 counts runs in binary expansion.
A275870 counts collapsible partitions, ranked by A300273.
A351014 counts distinct runs in standard compositions.
A353832 represents the operation of taking run-sums of a partition.
A353840-A353846 pertain to partition run-sum trajectory.
A353849 counts distinct run-sums in standard compositions.
Cf. A000041, A008284, A047966, A071625, A073093, A116608, A175413, A181819, A333755, A353848, A353867.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Total/@Split[#]&]],{n,0,15}]

a353837 = lambda n: sum( abs(BipartiteGraph( Matrix(len(p), len(D:=list(set.union(*map(lambda t: set(divisors(t)),p)))), lambda i,j: p[i]%D[j]==0) ).matching_polynomial()[len(D)-len(p)]) for p in Partitions(n,max_slope=-1) ) # Max Alekseyev, Sep 11 2023

A048768 Numbers n such that A048767(n) = n.

Original entry on oeis.org

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 11776, 14000, 19584, 21609, 28812, 29403, 29696, 43218, 43776, 44000, 58806, 63488, 75600, 96040, 104000, 105984, 123201, 126000

Offset: 1

Naohiro Nomoto

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions that are fixed points under the map described in A217605 (which interchanges the parts with their multiplicities). The enumeration of these partitions by sum is given by A217605. - Gus Wiseman, May 04 2019

			12 = (2^2)*(3^1) = (2nd prime)^pi(2) * (first prime)^pi(3).
From _Gus Wiseman_, May 04 2019: (Start)
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     9: {2,2}
    12: {1,1,2}
    18: {1,2,2}
    40: {1,1,1,3}
   112: {1,1,1,1,4}
   125: {3,3,3}
   250: {1,3,3,3}
   352: {1,1,1,1,1,5}
   360: {1,1,1,2,2,3}
   675: {2,2,2,3,3}
   832: {1,1,1,1,1,1,6}
  1008: {1,1,1,1,2,2,4}
  1125: {2,2,3,3,3}
  1350: {1,2,2,2,3,3}
  1500: {1,1,2,3,3,3}
  2176: {1,1,1,1,1,1,1,7}
  2250: {1,2,2,3,3,3}
  2401: {4,4,4,4}
(End)

Amiram Eldar, Table of n, a(n) for n = 1..128

A subsequence of A130091.

Cf. A008478, A048767, A048769, A056239, A098859, A109297, A109298, A112798, A118914, A217605, A325326, A325368.

wt[n_]:=Times@@Cases[FactorInteger[n],{p_,k_}:>Prime[k]^PrimePi[p]];
Select[Range[1000],wt[#]==#&] (* Gus Wiseman, May 04 2019 *)

is(n) = {my(f = factor(n), p = f[, 1], e = f[, 2]); #Set(e) == #e && prod(i = 1, #e, prime(e[i])^primepi(p[i])) == n;} \\ Amiram Eldar, Oct 20 2023

a(1) inserted and more terms added by Amiram Eldar, Oct 20 2023

A336866 Number of integer partitions of n without all distinct multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 15, 21, 28, 46, 56, 80, 114, 149, 192, 269, 337, 455, 584, 751, 943, 1234, 1527, 1944, 2422, 3042, 3739, 4699, 5722, 7100, 8668, 10634, 12880, 15790, 19012, 23093, 27776, 33528, 40102, 48264, 57469, 68793, 81727, 97372, 115227

Offset: 0

Gus Wiseman, Aug 09 2020

nonn

			The a(0) = 0 through a(9) = 15 partitions (empty columns shown as dots):
  .  .  .  (21)  (31)  (32)  (42)    (43)    (53)     (54)
                       (41)  (51)    (52)    (62)     (63)
                             (321)   (61)    (71)     (72)
                             (2211)  (421)   (431)    (81)
                                     (3211)  (521)    (432)
                                             (3221)   (531)
                                             (3311)   (621)
                                             (4211)   (3321)
                                             (32111)  (4221)
                                                      (4311)
                                                      (5211)
                                                      (32211)
                                                      (42111)
                                                      (222111)
                                                      (321111)

A098859 counts the complement.
A130092 gives the Heinz numbers of these partitions.
A001222 counts prime factors with multiplicity.
A013929 lists nonsquarefree numbers.
A047966 counts uniform partitions.
A047967 counts non-strict partitions.
A071625 counts distinct prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor with distinct prime multiplicities.
Cf. A000009, A000041, A008284, A116608, A118914, A124010, A242882, A255231, A325280, A325242, A329739, A336422, A336571.

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@Length/@Split[#]&]],{n,0,30}]

a(n) = A000041(n) - A098859(n).

A353838 Numbers whose prime indices have all distinct run-sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, May 23 2022

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 sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).

			The prime indices of 180 are {1,1,2,2,3}, with run-sums (2,4,3), so 180 is in the sequence.
The prime indices of 315 are {2,2,3,4}, with run-sums (4,3,4), so 315 is not in the sequence.

The version for all equal run-sums is A353833, counted by A304442.
These partitions are counted by A353837.
The complement is A353839.
The version for compositions is A353852, counted by A353850.
The greatest run-sum is given by A353862, least A353931.
The weak case is A353866, counted by A353864.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A098859 counts partitions with distinct multiplicities, ranked by A130091.
A165413 counts distinct run-sums in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A351014 counts distinct runs in standard compositions.
A353832 represents taking run-sums of a partition, compositions A353847.
A353840-A353846 pertain to partition run-sum trajectory.
Cf. A002110, A071625, A073093, A118914, A124010, A353834, A353861, A353867.

Select[Range[100],UnsameQ@@Cases[FactorInteger[#],{p_,k_}:>k*PrimePi[p]]&]

A328592 Numbers whose binary expansion has all different lengths of runs of 1's.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 11, 12, 13, 14, 15, 16, 19, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 35, 38, 39, 44, 46, 47, 48, 49, 50, 52, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 67, 70, 71, 76, 78, 79, 88, 92, 94, 95, 96, 97, 98, 100, 103, 104, 110, 111, 112, 113, 114

Offset: 1

Gus Wiseman, Oct 20 2019

nonn

Also numbers whose binary indices have different lengths of runs of successive parts. A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

The complement is {5, 9, 10, 17, 18, 20, 21, 27, ...}.

			The sequence of terms together with their binary expansions and binary indices begins:
   0:     0 ~ {}
   1:     1 ~ {1}
   2:    10 ~ {2}
   3:    11 ~ {1,2}
   4:   100 ~ {3}
   6:   110 ~ {2,3}
   7:   111 ~ {1,2,3}
   8:  1000 ~ {4}
  11:  1011 ~ {1,2,4}
  12:  1100 ~ {3,4}
  13:  1101 ~ {1,3,4}
  14:  1110 ~ {2,3,4}
  15:  1111 ~ {1,2,3,4}
  16: 10000 ~ {5}
  19: 10011 ~ {1,2,5}
  22: 10110 ~ {2,3,5}
  23: 10111 ~ {1,2,3,5}
  24: 11000 ~ {4,5}
  25: 11001 ~ {1,4,5}
  26: 11010 ~ {2,4,5}

The version for prime indices is A130091.
The binary expansion of n has A069010(n) runs of 1's.
The lengths of runs of 1's in the binary expansion of n are row n of A245563.
Numbers whose binary expansion has equal lengths of runs of 1's are A164707.
Cf. A000120, A003714, A014081, A098859, A121016, A328593, A328595, A328596.

Select[Range[0,100],UnsameQ@@Length/@Split[Join@@Position[Reverse[IntegerDigits[#,2]],1],#2==#1+1&]&]

A130092 Numbers with at least two factors having in their canonical prime factorization equal exponents.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 138, 140, 141, 142

Offset: 1

Reinhard Zumkeller, May 06 2007

nonn

Complement of A130091; A120944 and A085986 are subsequences;

a(n)>A130091(n) for n<=150, a(n) < A130091(n) for n>150.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Factorization

t[n_] := FactorInteger[n][[All, 2]];s = Select[Range[400], Union[t[#]] == Sort[t[#]] &] (* A130091 *)
Complement[Range[Max[s]], s]  (* A130092 *)
(* Clark Kimberling, Mar 12 2015 *)

cp's OEIS Frontend

A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

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A351295 Numbers whose multiset of prime factors has no permutation with all distinct run-lengths.

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A381432 Heinz numbers of section-sum partitions. Union of A381431.

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

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A353837 Number of integer partitions of n with all distinct run-sums.

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Sage

A048768 Numbers n such that A048767(n) = n.

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PARI

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A336866 Number of integer partitions of n without all distinct multiplicities.

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A353838 Numbers whose prime indices have all distinct run-sums.

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