A351013 -id:A351013 - cp's OEIS frontend

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 4, 4, 1, 0, 4, 4, 0, 0, 1, 6, 1, 0, 4, 6, 4, 0, 1, 6, 4, 0, 1, 6, 1, 0, 0, 8, 1, 0, 4, 0, 6, 0, 1, 0, 6, 0, 6, 8, 1, 0, 1, 10, 0, 0, 8, 6, 1, 0, 8, 6, 1, 0, 1, 10, 0, 0, 6, 6, 1, 0, 0, 12, 1, 0, 16

Offset: 1

Gus Wiseman, Apr 09 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}.

			The a(n) partitions for n = 6, 21, 30, 46:
  (1,1,2)  (1,1,1,1,2,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,2,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,2,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,2,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of 1 are A008578.
For anti-run permutations we have A335125.
For just prime indices we have A382771, firsts A382772, equal A382857.
These permutations for factorials are counted by A382774, equal A335407.
For equal instead of distinct run-lengths we have A382858.
Positions of 0 are A382912, complement A382913.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000670, A000720, A003242, A048767, A130091, A181821, A238130, A305936, A351013, A351202, A382876, A382879.

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

a(n) = A382771(A181821(n)) = A382771(A304660(n)).

A353400 Number of integer compositions of n with all run-lengths > 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 2, 4, 4, 5, 11, 11, 14, 27, 29, 37, 61, 72, 97, 147, 181, 246, 368, 470, 632, 914, 1198, 1611, 2286, 3018, 4079, 5709, 7619, 10329, 14333, 19258, 26142, 36069, 48688, 66114, 90800, 122913, 167020, 228735, 310167, 421708, 576499, 782803

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(7) = 1 through a(12) = 11 compositions:
  1111111   2222       333         22222        1112222       444
            11111111   111222      1111222      2222111       3333
                       222111      2221111      11111222      111333
                       111111111   1111111111   22211111      222222
                                                11111111111   333111
                                                              11112222
                                                              22221111
                                                              111111222
                                                              111222111
                                                              222111111
                                                              111111111111

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The = 2 version is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A078012, both A353428.
The version for partitions is A100405.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A274174 counts compositions with equal parts contiguous.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A098859, A165413, A175413, A333755, A351013, A353390, A353391, A353401.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=3..n/i), 0), i=1..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(21)-a(49) from Alois P. Heinz, May 17 2022

A353392 Number of compositions of n whose own run-lengths are a consecutive subsequence.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 2, 2, 2, 8, 12, 16, 20, 35, 46, 59, 81, 109, 144, 202, 282

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 0 through a(10) = 12 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  1  .  .  22  122  1122  11221  21122  333     1333
                  221  2211  12211  22112  22113   2233
                                           22122   3322
                                           31122   3331
                                           121122  22114
                                           122112  41122
                                           211221  122113
                                           221121  131122
                                                   221131
                                                   311221
                                                   1211221
                                                   1221121

The non-consecutive version for partitions is A325702.
The non-consecutive version is A353390, ranked by A353402.
The non-consecutive recursive version is A353391, ranked by A353431.
The non-consecutive reverse version is A353403.
The recursive version is A353430.
These compositions are ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A008965, A032020, A103295, A103300, A114901, A238279, A324572, A325705, A333224, A333755, A351013, A353401.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],#=={}||MemberQ[Join@@Table[Take[#,{i,j}],{i,Length[#]},{j,i,Length[#]}],Length/@Split[#]]&]],{n,0,15}]

A353401 Number of integer compositions of n with all prime run-lengths.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 4, 3, 6, 9, 10, 18, 27, 35, 54, 83, 107, 176, 242, 354, 515, 774, 1070, 1648, 2332, 3429, 4984, 7326, 10521, 15591, 22517, 32908, 48048, 70044, 101903, 149081, 216973, 316289, 461959, 672664, 981356, 1431256, 2086901, 3041577, 4439226, 6467735

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(9) = 9 compositions (empty column indicated by dot, 0 is the empty composition):
  0   .  11   111   22   11111   33     11122     44       333
                                 222    22111     1133     11133
                                 1122   1111111   3311     33111
                                 2211             11222    111222
                                                  22211    222111
                                                  112211   1111122
                                                           1112211
                                                           1122111
                                                           2211111

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

The case of runs equal to 2 is A003242 aerated.
The <= 1 version is A003242 ranked by A333489.
The version for parts instead of run-lengths is A023360, both A353429.
The version for partitions is A055923.
The > 1 version is A114901, ranked by A353427.
The <= 2 version is A128695, matching A335464.
The > 2 version is A353400, partitions A100405.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351013, A351017.
A005811 counts runs in binary expansion.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A167606 counts compositions with adjacent parts coprime.
A329738 counts uniform compositions, partitions A047966.
Cf. A078012, A165413, A175413, A274174, A333381, A333755, A353390, A353391, A353392, A353402, A353403.

b:= proc(n, h) option remember; `if`(n=0, 1, add(`if`(i<>h, add(
     `if`(isprime(j), b(n-i*j, i), 0), j=2..n/i), 0), i=1..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Length/@Split[#],_?(!PrimeQ[#]&)]&]],{n,0,15}]

a(21)-a(45) from Alois P. Heinz, May 18 2022

A353403 Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).

Original entry on oeis.org

1, 1, 0, 0, 3, 2, 5, 12, 16, 30, 45, 94, 159, 285, 477, 864, 1487, 2643

Offset: 0

Gus Wiseman, May 15 2022

			The a(0) = 1 through a(7) = 12 compositions:
  ()  (1)  .  .  (22)   (1121)  (1113)  (1123)
                 (112)  (1211)  (1122)  (1132)
                 (211)          (1221)  (2311)
                                (2211)  (3211)
                                (3111)  (11131)
                                        (11212)
                                        (11221)
                                        (12112)
                                        (12211)
                                        (13111)
                                        (21121)
                                        (21211)

The non-reversed version is A353390, ranked by A353402, partitions A325702.
The non-reversed recursive version is A353391, ranked by A353431.
The non-reversed consecutive case is A353392, ranked by A353432.
The non-reversed recursive consecutive version is A353430.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223, partitions A108917.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
Cf. A005811, A032020, A103295, A114640, A165413, A324572, A333755, A353400, A353401, A353426.

Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n],MemberQ[Subsets[#],Reverse[Length/@Split[#]]]&]],{n,0,15}]

A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45

Offset: 0

Gus Wiseman, May 16 2022

			The a(n) compositions for selected n (A..E = 10..14):
  n=4:  n=6:    n=9:      n=10:     n=12:     n=14:
-----------------------------------------------------------
  (4)   (6)     (9)       (A)       (C)       (E)
  (22)  (1122)  (333)     (2233)    (2244)    (2255)
        (2211)  (121122)  (3322)    (4422)    (5522)
                (221121)  (131122)  (151122)  (171122)
                          (221131)  (221124)  (221126)
                                    (221142)  (221135)
                                    (221151)  (221153)
                                    (241122)  (221162)
                                    (421122)  (221171)
                                              (261122)
                                              (351122)
                                              (531122)
                                              (621122)
                                              (122121122)
                                              (221121221)

Non-recursive non-consecutive version: counted by A353390, ranked by A353402, reverse A353403, partitions A325702.
Non-consecutive version: A353391, ranked by A353431, partitions A353426.
Non-recursive version: A353392, ranked by A353432.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A329738 counts uniform compositions, partitions A047966.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],yoyQ]],{n,0,15}]

A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).

Original entry on oeis.org

1, 6, 24, 30, 36, 180, 210, 360, 420, 720, 1080, 1260, 1800, 2160, 2310, 2520, 3600, 4620, 5040, 5400, 6300, 7560, 10800, 12600, 13860, 15120, 21600, 25200, 25920, 27000, 27720, 30030, 32400, 37800, 44100, 45360, 46656, 50400, 54000, 55440, 60060, 60480, 64800

Offset: 1

Gus Wiseman, Apr 09 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, sum A056239.

			The permutations for n = 6, 720, 36, 25920, 30:
  (1,2)  (1,2,1,2,1,3,1)  (1,1,2,2)  (1,2,1,2,1,2,1,2,1,3,1)  (1,2,3)
  (2,1)  (1,2,1,3,1,2,1)  (1,2,1,2)  (1,2,1,2,1,2,1,3,1,2,1)  (1,3,2)
         (1,3,1,2,1,2,1)  (2,1,2,1)  (1,2,1,2,1,3,1,2,1,2,1)  (2,1,3)
                          (2,2,1,1)  (1,2,1,3,1,2,1,2,1,2,1)  (2,3,1)
                                     (1,3,1,2,1,2,1,2,1,2,1)  (3,1,2)
                                                              (3,2,1)
The terms together with their prime indices begin:
      1: {}
      6: {1,2}
     24: {1,1,1,2}
     30: {1,2,3}
     36: {1,1,2,2}
    180: {1,1,2,2,3}
    210: {1,2,3,4}
    360: {1,1,1,2,2,3}
    420: {1,1,2,3,4}
    720: {1,1,1,1,2,2,3}
   1080: {1,1,1,2,2,2,3}
   1260: {1,1,2,2,3,4}
   1800: {1,1,1,2,2,3,3}
   2160: {1,1,1,1,2,2,2,3}
   2310: {1,2,3,4,5}
   2520: {1,1,1,2,2,3,4}
   3600: {1,1,1,1,2,2,3,3}

Positions of first appearances in A382857 (zeros A382879), by signature A382858.
For distinct run-lengths we have A382772, firsts of A382771 (by signature A382773).
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A098859, A130091, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],SameQ@@Length/@Split[#]&]],{n,0,1000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

Original entry on oeis.org

10, 14, 22, 26, 28, 33, 34, 38, 39, 44, 46, 51, 52, 55, 57, 58, 62, 66, 68, 69, 74, 76, 78, 82, 85, 86, 87, 88, 92, 93, 94, 95, 102, 104, 106, 111, 114, 115, 116, 118, 119, 122, 123, 124, 129, 130, 134, 136, 138, 141, 142, 145, 146, 148, 152, 153, 155, 156

Offset: 1

Gus Wiseman, Apr 09 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, sum A056239.

			The terms together with their prime indices begin:
  10: {1,3}
  14: {1,4}
  22: {1,5}
  26: {1,6}
  28: {1,1,4}
  33: {2,5}
  34: {1,7}
  38: {1,8}
  39: {2,6}
  44: {1,1,5}
  46: {1,9}
  51: {2,7}
  52: {1,1,6}
  55: {3,5}
  57: {2,8}
  58: {1,10}
  62: {1,11}
  66: {1,2,5}

For anti-run permutations we have A335126, complement A335127.
Zeros of A382858, anti-run A335125.
For prime indices instead of signature we have A382879, counted by A382915.
For distinct run-lengths we have A382912 (zeros of A382773), complement A382913.
A003963 gives product of prime indices.
A056239 adds up prime indices, row sums of A112798.
A140690 lists numbers whose binary expansion has equal run-lengths, distinct A044813.
A304442 counts partitions with equal run-sums, ranks A353833.
A164707 lists numbers whose binary form has equal runs of ones, distinct A328592.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
Cf. A382857 (firsts A382878), A382771 (firsts A382772).
Cf. A000720, A000961, A001221, A001222, A048767, A181821, A305936, A351013, A382877.

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

A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

Original entry on oeis.org

1, 6, 12, 96, 360, 1536, 3456, 5184, 5760, 6144, 7776, 13824, 23040, 24576, 55296, 62208, 92160

Offset: 1

Gus Wiseman, Apr 09 2025

			The permutations for n = 12, 96, 360, 1536:
  (1,1,2)  (1,1,1,1,1,2)  (1,1,1,2,2,3)  (1,1,1,1,1,1,1,1,1,2)
  (2,1,1)  (1,1,1,2,1,1)  (1,1,1,3,2,2)  (1,1,1,1,1,1,1,2,1,1)
           (1,1,2,1,1,1)  (2,2,1,1,1,3)  (1,1,1,1,1,1,2,1,1,1)
           (2,1,1,1,1,1)  (2,2,3,1,1,1)  (1,1,1,1,1,2,1,1,1,1)
                          (3,1,1,1,2,2)  (1,1,1,1,2,1,1,1,1,1)
                          (3,2,2,1,1,1)  (1,1,1,2,1,1,1,1,1,1)
                                         (1,1,2,1,1,1,1,1,1,1)
                                         (2,1,1,1,1,1,1,1,1,1)

Positions of first appearances in A382771, by signature A382773.
For equal run-lengths we have A382878, firsts of A382857, zeros A382879.
A044813 lists numbers whose binary expansion has distinct run-lengths, equal A140690.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A098859 counts partitions with distinct multiplicities, ordered A242882.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A328592 lists numbers whose binary form has distinct runs of ones, equal A164707.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A000720, A001221, A001222, A003242, A048767, A130091, A238130, A305936, A351013, A351202, A382876.

y=Table[Length[Select[Permutations[Join@@ConstantArray@@@FactorInteger[n]],UnsameQ@@Length/@Split[#]&]],{n,0,100000}];
fip[y_]:=Select[Range[Length[y]],!MemberQ[Take[y,#-1],y[[#]]]&];
fip[Rest[y]]

A165933 Least integer, k, whose value is n in A165413.

Original entry on oeis.org

1, 4, 35, 536, 16775, 1060976, 135007759, 34460631520, 17617985239071, 18027600169142208, 36907002795598798911, 151143401509104346210176, 1238053384151947477501575295, 20283338091738780737237428502272, 664629209970464486086782992577855743

Offset: 1

Robert G. Wilson v, Sep 30 2009

An alternative name: The smallest number whose binary expansion has exactly n distinct run-lengths. - Gus Wiseman, Feb 21 2022

Term a(n) has one 1, followed by n 0's, then two 1's, (n-1) 0's, ..., up to n runs; see Python program. - Michael S. Branicky, Feb 22 2022

			a(1) in binary is 1, a(2) in binary is 100, a(3) in binary is 100011, a(4) in binary is 1000011000, etc.
From _Gus Wiseman_, Feb 21 2022: (Start)
The terms and their binary expansions begin:
  n              a(n)
  1:               1 =                                             1
  2:               4 =                                           100
  3:              35 =                                        100011
  4:             536 =                                    1000011000
  5:           16775 =                               100000110000111
  6:         1060976 =                         100000011000001110000
  7:       135007759 =                  1000000011000000111000001111
  8:     34460631520 =          100000000110000000111000000111100000
  9:  17617985239071 = 100000000011000000001110000000111100000011111
(End)

Michael S. Branicky, Table of n, a(n) for n = 1..81

A subset of A044813 (distinct run-lengths) and of A175413 (distinct runs).
These are the positions of first appearances in A165413.
The version for runs instead of run-lengths is A350952, firsts of A297770.
A000120 counts binary weight.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A329739 = compositions, for runs A351013.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A003242, A070939, A085207, A098859, A325545, A328592, A351015, A351202, A351203, A351291.

g[n_] := Table[ {Table[1, {i}], Table[0, {n - i + 1}]}, {i, Floor[(n + If[ OddQ@n, 1, 0])/2]}]; f[n_] := FromDigits[ If[ OddQ@n, Flatten@ Most@ Flatten[ g@n, 1], Flatten@ g@n], 2]; Array[f, 14]
s=Table[Length[Union[Length/@Split[IntegerDigits[n,2]]]],{n,0,1000}]; Table[Position[s,k][[1,1]]-1,{k,Union[s]}] (* Gus Wiseman, Feb 21 2022 *)

def a(n): # returns term by construction
    if n == 1: return 1
    q, r = divmod(n+1, 2)
    s = "".join("1"*i + "0"*(n+1-i) for i in range(1, q+1))
    if r == 0: s = s.rstrip("0")
    return int(s, 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 22 2022

a(15) and beyond from Michael S. Branicky, Feb 22 2022

cp's OEIS Frontend

A382773 Number of ways to permute a multiset whose multiplicities are the prime indices of n so that the run-lengths are all different.

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A353400 Number of integer compositions of n with all run-lengths > 2.

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A353392 Number of compositions of n whose own run-lengths are a consecutive subsequence.

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A353401 Number of integer compositions of n with all prime run-lengths.

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A353403 Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).

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A353430 Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

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A382878 Set of positions of first appearances in A382857 (permutations of prime indices with equal run-lengths).

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A382914 Numbers k such that it is not possible to permute a multiset whose multiplicities are the prime indices of k so that the run-lengths are all equal.

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A382772 Set of positions of first appearances in A382771 (permutations of prime indices with distinct run-lengths).

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