A114640 -id:A114640 - cp's OEIS frontend

A109297 Primal codes of finite permutations on positive integers.

1, 2, 9, 12, 18, 40, 112, 125, 250, 352, 360, 540, 600, 675, 832, 1008, 1125, 1350, 1500, 2176, 2250, 2268, 2352, 2401, 3168, 3969, 4802, 4864, 7488, 7938, 10692, 11616, 11776, 14000, 19584, 21609, 27440, 28812, 29403, 29696, 32448, 35000, 37908, 43218, 43776

Offset: 1

Jon Awbrey, Jul 08 2005

nonn

A finite permutation is a bijective mapping from a finite set to itself, counting the empty mapping as a permutation of the empty set.

Also Heinz numbers of integer partitions where the set of distinct parts is equal to the set of distinct multiplicities. These partitions are counted by A114640. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). - Gus Wiseman, Apr 02 2019

			Writing (prime(i))^j as i:j, we have the following table:
Primal Codes of Finite Permutations on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = 2:2
` ` `12 = 1:2 2:1
` ` `18 = 1:1 2:2
` ` `40 = 1:3 3:1
` ` 112 = 1:4 4:1
` ` 125 = 3:3
` ` 250 = 1:1 3:3
` ` 352 = 1:5 5:1
` ` 360 = 1:3 2:2 3:1
` ` 540 = 1:2 2:3 3:1
` ` 600 = 1:3 2:1 3:2
` ` 675 = 2:3 3:2
` ` 832 = 1:6 6:1
` `1008 = 1:4 2:2 4:1
` `1125 = 2:2 3:3
` `1350 = 1:1 2:3 3:2
` `1500 = 1:2 2:1 3:3
` `2176 = 1:7 7:1
` `2250 = 1:1 2:2 3:3

Amiram Eldar, Table of n, a(n) for n = 1..300 (terms 1..100 from Alois P. Heinz)
Jon Awbrey, Riffs and Rotes.

Subsequence of A130091.
Cf. A076954, A106177, A108352, A108371, A109298, A109301, A056239, A112798, A114640, A118914.
Cf. A324524, A324525, A324571, A325127, A325128, A325130, A325131.

a:= proc(n) option remember; local k; for k from 1+`if`(n=1, 0,
      a(n-1)) while (l-> sort(map(i-> i[2], l)) <> sort(map(
      i-> numtheory[pi](i[1]), l)))(ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..45);  # Alois P. Heinz, Mar 08 2019

Select[Range[1000],#==1||Union[PrimePi/@First/@FactorInteger[#]]==Union[Last/@FactorInteger[#]]&] (* Gus Wiseman, Apr 02 2019 *)

is(n) = {my(f = factor(n), p = f[,1], e = vecsort(f[,2])); for(i=1, #p, if(primepi(p[i]) != e[i], return(0))); 1}; \\ Amiram Eldar, Jul 30 2022

More terms from Franklin T. Adams-Watters, Dec 19 2005

Offset set to 1 by Alois P. Heinz, Mar 08 2019

A114639 Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.

Original entry on oeis.org

1, 0, 2, 2, 2, 3, 5, 4, 7, 7, 13, 16, 19, 23, 33, 34, 44, 58, 63, 80, 101, 112, 139, 171, 196, 234, 288, 328, 394, 478, 545, 658, 777, 881, 1050, 1236, 1414, 1666, 1936, 2216, 2592, 3018, 3428, 3992, 4604, 5243, 6069, 6986, 7951, 9139, 10447, 11892, 13625

Offset: 0

Vladeta Jovovic, Feb 18 2006

nonn

The Heinz numbers of these partitions are given by A325131. - Gus Wiseman, Apr 02 2019

			From _Gus Wiseman_, Apr 02 2019: (Start)
The a(2) = 2 through a(9) = 7 partitions:
  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
  (11)  (111)  (1111)  (32)     (33)      (43)       (44)        (54)
                       (11111)  (42)      (52)       (53)        (63)
                                (222)     (1111111)  (62)        (72)
                                (111111)             (2222)      (432)
                                                     (3311)      (222111)
                                                     (11111111)  (111111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A052335, A087153, A114640, A115584, A117144, A276429, A324572, A325130, A325131, A336032.

b:= proc(n, i, p, m) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1, p, select(x-> x x<=n-i*j, p union {i}),
         select(x-> x b(n$2, {}$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 09 2016

b[n_, i_, p_, m_] := b[n, i, p, m] = If[n == 0, 1, If[i<1, 0, b[n, i-1, p, Select[m, #Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Intersection[#,Length/@Split[#]]=={}&]],{n,0,30}] (* Gus Wiseman, Apr 02 2019 *)

a(0)=1 prepended and more terms from Alois P. Heinz, Aug 09 2016

A325705 Number of integer partitions of n containing all of their distinct multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 3, 2, 4, 3, 7, 8, 16, 15, 24, 28, 39, 44, 68, 80, 98, 130, 167, 200, 259, 320, 396, 497, 601, 737, 910, 1107, 1335, 1631, 1983, 2372, 2887, 3439, 4166, 4949, 5940, 7043, 8450, 9980, 11884, 13984, 16679, 19493, 23162, 27050, 31937, 37334, 43926

Offset: 0

Gus Wiseman, May 18 2019

nonn

The Heinz numbers of these partitions are given by A325706.

			The partition (4,2,1,1,1,1) has distinct multiplicities {1,4}, both of which belong to the partition, so it is counted under a(10).
The a(0) = 1 through a(10) = 16 partitions:
  ()  (1)  (21)  (22)   (41)   (51)    (61)    (71)     (81)     (91)
                 (31)   (221)  (321)   (421)   (431)    (333)    (541)
                 (211)         (2211)  (3211)  (521)    (531)    (631)
                               (3111)          (3221)   (621)    (721)
                                               (4211)   (3321)   (3322)
                                               (32111)  (4221)   (3331)
                                               (41111)  (5211)   (4321)
                                                        (32211)  (5221)
                                                                 (6211)
                                                                 (32221)
                                                                 (33211)
                                                                 (42211)
                                                                 (43111)
                                                                 (322111)
                                                                 (421111)
                                                                 (511111)

Cf. A109297, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325706, A325707, A325755.

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

A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a 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, 22, 38, 45, 87, 93

Offset: 0

Gus Wiseman, May 15 2022

			The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

The non-recursive version is A353390, ranked by A353402.
The non-recursive consecutive version is A353392, ranked by A353432.
The non-recursive reverse version is A353403.
The unordered version is A353426, ranked by A353393 (nonprime A353389).
The consecutive version is A353430.
These compositions are ranked by A353431.
A003242 counts anti-run compositions, ranked by A333489.
A011782 counts compositions.
A329738 counts uniform compositions, partitions A047966.
A114901 counts compositions with no runs of length 1.
A169942 counts Golomb rulers, ranked by A333222.
A325676 counts knapsack compositions, ranked by A333223.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

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

A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

Original entry on oeis.org

0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466

Offset: 0

Gus Wiseman, May 15 2022

nonn

First differs from A353432 (the consecutive case) in having 0 and 53.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The initial terms, their binary expansions, and the corresponding standard compositions:
    0:          0  ()
    1:          1  (1)
   10:       1010  (2,2)
   21:      10101  (2,2,1)
   26:      11010  (1,2,2)
   43:     101011  (2,2,1,1)
   53:     110101  (1,2,2,1)
   58:     111010  (1,1,2,2)
  107:    1101011  (1,2,2,1,1)
  117:    1110101  (1,1,2,2,1)
  174:   10101110  (2,2,1,1,2)
  186:   10111010  (2,1,1,2,2)
  292:  100100100  (3,3,3)
  314:  100111010  (3,1,1,2,2)
  346:  101011010  (2,2,1,2,2)
  348:  101011100  (2,2,1,1,3)
  349:  101011101  (2,2,1,1,2,1)
  373:  101110101  (2,1,1,2,2,1)
  430:  110101110  (1,2,2,1,1,2)
  442:  110111010  (1,2,1,1,2,2)

The version for partitions is A325755, counted by A325702.
These compositions are counted by A353390.
The recursive version is A353431, counted by A353391.
The consecutive case is A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, reverse A228351.
A333769 lists run-lengths of compositions in standard order.
Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, consecutive A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
- Runs are A272919.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];
Select[Range[0,100],rosQ[stc[#]]&]

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}]

A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 16 2022

nonn

First differs from A353696 (the consecutive version) in having 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are subsequence but not a consecutive subsequence.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The initial terms, their binary expansions, and the corresponding standard compositions:
     0:           0  ()
     1:           1  (1)
     2:          10  (2)
     4:         100  (3)
     8:        1000  (4)
    10:        1010  (2,2)
    16:       10000  (5)
    32:      100000  (6)
    43:      101011  (2,2,1,1)
    58:      111010  (1,1,2,2)
    64:     1000000  (7)
   128:    10000000  (8)
   256:   100000000  (9)
   292:   100100100  (3,3,3)
   349:   101011101  (2,2,1,1,2,1)
   442:   110111010  (1,2,1,1,2,2)
   512:  1000000000  (10)
   586:  1001001010  (3,3,2,2)
   676:  1010100100  (2,2,3,3)
   697:  1010111001  (2,2,1,1,3,1)

The non-recursive version for partitions is A325755, counted by A325702.
These compositions are counted by A353391.
The version for partitions A353393, counted by A353426, w/o primes A353389.
The non-recursive version is A353402, counted by A353390.
The non-recursive consecutive case is A353432, counted by A353392.
The consecutive case is A353696, counted by A353430.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, rev A228351, run-lens A333769.
A329738 counts uniform compositions, partitions A047966.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, multisets A225620, strict A333255, sets A333256.
- Constant compositions are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Knapsack compositions are A333223, counted by A325676.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&& rorQ[Length/@Split[y]];
Select[Range[0,100],rorQ[stc[#]]&]

A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

Original entry on oeis.org

1, 2, 5, 8, 10, 11, 17, 22, 23, 31, 32, 34, 40, 41, 46, 47, 55, 59, 62, 67, 73, 82, 83, 85, 88, 94, 97, 103, 109, 110, 115, 118, 125, 127, 128, 134, 136, 137, 146, 149, 155, 157, 160, 166, 167, 170, 179, 184, 187, 191, 194, 197, 205, 206, 211, 218, 227, 230

Offset: 1

Gus Wiseman, Mar 18 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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions with all odd parts and all odd multiplicities, counted by A117958.

			The terms together with their prime indices begin:
   1 = 1
   2 = prime(1)
   5 = prime(3)
   8 = prime(1)^3
  10 = prime(1) prime(3)
  11 = prime(5)
  17 = prime(7)
  22 = prime(1) prime(5)
  23 = prime(9)
  31 = prime(11)
  32 = prime(1)^5
  34 = prime(1) prime(7)
  40 = prime(1)^3 prime(3)

David A. Corneth, Table of n, a(n) for n = 1..10000

The restriction to primes is A031368.
The first condition alone is A066208, counted by A000009.
These partitions are counted by A117958.
The squarefree case is A258116, even A258117.
The second condition alone is A268335, counted by A055922.
The even-even version is A352141 counted by A035444.
A000290 = exponents all even, counted by A035363.
A056166 = exponents all prime, counted by A055923.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162641 counts even prime exponents, odd A162642.
A257991 counts odd prime indices, even A257992.
A325131 = disjoint indices from exponents, counted by A114639.
A346068 = indices and exponents all prime, counted by A351982.
A351979 = odd indices with even exponents, counted by A035457.
A352140 = even indices with odd exponents, counted by A055922 aerated.
A352143 = odd indices with odd conjugate indices, counted by A053253 aerated.
Cf. A000720, A028260, A055396, A061395, A106529, A181819, A195017, A241638, A276078, A324517, A324524, A324525, A325698, A325700.

Select[Range[100],#==1||And@@OddQ/@PrimePi/@First/@FactorInteger[#]&&And@@OddQ/@Last/@FactorInteger[#]&]

from itertools import count, islice
from sympy import primepi, factorint
def A352142_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:all(map(lambda x:x[1]%2 and primepi(x[0])%2, factorint(k).items())),count(max(startvalue,1)))
A352142_list = list(islice(A352142_gen(),30)) # Chai Wah Wu, Mar 18 2022

Intersection of A066208 and A268335.
A257991(a(n)) = A001222(a(n)).
A162642(a(n)) = A001221(a(n)).
A257992(a(n)) = A162641(a(n)) = 0.

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}]

A325706 Heinz numbers of integer partitions containing all of their distinct multiplicities.

Original entry on oeis.org

1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 36, 38, 40, 42, 46, 58, 60, 62, 66, 70, 74, 78, 82, 84, 86, 90, 94, 102, 106, 110, 112, 114, 118, 120, 122, 125, 126, 130, 132, 134, 138, 142, 146, 150, 154, 156, 158, 166, 170, 174, 178, 180, 182, 186, 190, 194, 198

Offset: 1

Gus Wiseman, May 18 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Also numbers n divisible by the squarefree kernel of their "shadow" A181819(n).
The enumeration of these partitions by sum is given by A325705.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    6: {1,2}
    9: {2,2}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   18: {1,2,2}
   22: {1,5}
   26: {1,6}
   30: {1,2,3}
   34: {1,7}
   36: {1,1,2,2}
   38: {1,8}
   40: {1,1,1,3}
   42: {1,2,4}
   46: {1,9}
   58: {1,10}
   60: {1,1,2,3}
   62: {1,11}

Cf. A056239, A109297, A112798, A114639, A114640, A181819, A225486, A290689, A324753, A324843, A325702, A325705, A325707, A325755.

Select[Range[100],#==1||SubsetQ[PrimePi/@First/@FactorInteger[#],Last/@FactorInteger[#]]&]

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A109297 Primal codes of finite permutations on positive integers.

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A114639 Number of partitions of n such that the set of parts and the set of multiplicities of parts are disjoint.

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A325705 Number of integer partitions of n containing all of their distinct multiplicities.

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A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

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A353402 Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

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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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A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

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A352142 Numbers whose prime factorization has all odd indices and all odd exponents.

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