A329131 -id:A329131 - cp's OEIS frontend

A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

1, 2, 12, 112, 122, 1112, 1122, 1222, 11112, 11122, 11212, 11222, 12122, 12222, 111112, 111122, 111212, 111222, 112122, 112212, 112222, 121222, 122222, 1111112, 1111122, 1111212, 1111222, 1112112, 1112122, 1112212, 1112222, 1121122

Offset: 1

N. J. A. Sloane, Feb 03 2005

A Lyndon word is primitive (not a power of another word) and is earlier in lexicographic order than any of its cyclic shifts.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. Bassino, J. Clement and C. Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Math. 290 (2005), 1-25.
Émilie Charlier, Manon Philibert, Manon Stipulanti, Nyldon words, arXiv:1804.09735 [math.CO], 2018. See Table 1.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012. - From _N. J. A. Sloane_, Dec 31 2012
Wikipedia, Lyndon word
Reinhard Zumkeller, Haskell programs for some sequences concerning Lyndon words
Index entries for sequences related to Lyndon words

Cf. A001037, A074650, A102660, A210584, A210585.
The "co" version is A329318.
A triangular version is A296657.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is Lyndon are A328596.
Length of the Lyndon factorization of the binary expansion is A211100.
Cf. A059966, A060223, A275692, A281013, A296373, A329131, A329313.

Haskell
```
cf. link.
```

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
Join@@Table[FromDigits/@Select[Tuples[{1,2},n],lynQ],{n,5}] (* Gus Wiseman, Nov 14 2019 *)

is_A102659(n)={ vecsort(d=digits(n))!=d&&for(i=1,#d-1, n>[1,10^(#d-i)]*divrem(n,10^i)&&return); fordiv(#d,L,L<#d && d==concat(Col(vector(#d/L,i,1)~*vecextract(d,2^L-1))~)&&return); !setminus(Set(d),[1,2])} \\ The last check is the least expensive one, but not useful if we test only numbers with digits {1,2}.
for(n=1,6,p=vector(n,i,10^(n-i))~;forvec(d=vector(n,i,[1,2]),is_A102659(m=d*p)&&print1(m","))) \\ One could use is_A102660 instead of is_A102659 here. - M. F. Hasler, Mar 08 2014

A102659 = A102660 intersect A007931 = A213969 intersect A239016. - M. F. Hasler, Mar 10 2014

More terms from Franklin T. Adams-Watters, Dec 14 2006

Definition improved by Reinhard Zumkeller, Mar 23 2012

A211100 Number of factors in Lyndon factorization of binary expansion of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 2, 3, 2, 6, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 7, 3, 4, 3, 5, 3, 4, 3, 6, 4, 5, 3, 5, 4, 4, 3, 7, 4, 5, 4, 6, 5, 5, 4, 7

Offset: 0

N. J. A. Sloane, Mar 31 2012

nonn

Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). a(n) = number of factors in Lyndon factorization of binary expansion of n.
It appears that a(n) = k for the first time when n = 2^(k-1)+1.
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. - Gus Wiseman, Nov 12 2019

			n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors, so a(25) = 3.
Here are the Lyndon factorizations for small values of n:
.0.
.1.
.1.0.
.1.1.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.1.0.0.0.
.1.001.
.1.01.0.
.1.011.
.1.1.0.0.
...

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Maple programs for A211100 etc.

Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof).
A211095 and A211096 give information about the smallest (or rightmost) factor. Cf. A211097, A211098, A211099.
Row-lengths of A329314.
The "co-" version is A329312.
Positions of 2's are A329327.
The reversed version is A329313.
The inverted version is A329312.
Ignoring the first digit gives A211097.
Cf. A059966, A060223, A275692, A296658, A328594, A328595, A328596, A329131, A329325, A329326.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
Table[Length[lynfac[IntegerDigits[n,2]]],{n,0,30}] (* Gus Wiseman, Nov 12 2019 *)

A329312 Length of the co-Lyndon factorization of the binary expansion of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 2, 3, 2, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3

Offset: 1

Gus Wiseman, Nov 10 2019

nonn

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

Also the length of the Lyndon factorization of the inverted binary expansion of n, where the inverted digits are 1 minus the binary digits.

			The binary indices of 1..20 together with their co-Lyndon factorizations are:
   1:     (1) = (1)
   2:    (10) = (10)
   3:    (11) = (1)(1)
   4:   (100) = (100)
   5:   (101) = (10)(1)
   6:   (110) = (110)
   7:   (111) = (1)(1)(1)
   8:  (1000) = (1000)
   9:  (1001) = (100)(1)
  10:  (1010) = (10)(10)
  11:  (1011) = (10)(1)(1)
  12:  (1100) = (1100)
  13:  (1101) = (110)(1)
  14:  (1110) = (1110)
  15:  (1111) = (1)(1)(1)(1)
  16: (10000) = (10000)
  17: (10001) = (1000)(1)
  18: (10010) = (100)(10)
  19: (10011) = (100)(1)(1)
  20: (10100) = (10100)

The non-"co" version is A211100.
Positions of 1's are A275692.
The reversed version is A329326.
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A329131, A329314, A329318, A329324, A329325.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[colynfac[IntegerDigits[n,2]]],{n,100}]

A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 1, 3, 2, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3

Offset: 0

Gus Wiseman, Nov 11 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).

			The sequence of reversed binary expansions of the nonnegative integers together with their Lyndon factorizations begins:
   0:      () = ()
   1:     (1) = (1)
   2:    (01) = (01)
   3:    (11) = (1)(1)
   4:   (001) = (001)
   5:   (101) = (1)(01)
   6:   (011) = (011)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0001)
   9:  (1001) = (1)(001)
  10:  (0101) = (01)(01)
  11:  (1101) = (1)(1)(01)
  12:  (0011) = (0011)
  13:  (1011) = (1)(011)
  14:  (0111) = (0111)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (00001)
  17: (10001) = (1)(0001)
  18: (01001) = (01)(001)
  19: (11001) = (1)(1)(001)
  20: (00101) = (00101)

The non-reversed version is A211100.
Positions of 1's are A328596.
The "co" version is A329326.
Binary Lyndon words are counted by A001037 and ranked by A102659.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose reversed binary expansion is a aperiodic are A328594.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A027375, A059966, A060223, A121016, A211097, A275692, A329131, A329314,  A329317, A329325.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#1]]&]]]];
Table[If[n==0,0,Length[lynfac[Reverse[IntegerDigits[n,2]]]]],{n,0,30}]

A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 3, 3, 2, 6, 3, 4, 2, 5, 4, 3, 2

Offset: 1

Gus Wiseman, Nov 11 2019

nonn

First differs from A211100 at a(77) = 3, A211100(77) = 2. The reversed binary expansion of 77 is (1011001), with co-Lyndon factorization (10)(1100)(1), while the binary expansion is (1001101), with Lyndon factorization of (1)(001101).

The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).

			The reversed binary expansion of each positive integer together with their co-Lyndon factorizations begins:
   1:     (1) = (1)
   2:    (01) = (0)(1)
   3:    (11) = (1)(1)
   4:   (001) = (0)(0)(1)
   5:   (101) = (10)(1)
   6:   (011) = (0)(1)(1)
   7:   (111) = (1)(1)(1)
   8:  (0001) = (0)(0)(0)(1)
   9:  (1001) = (100)(1)
  10:  (0101) = (0)(10)(1)
  11:  (1101) = (110)(1)
  12:  (0011) = (0)(0)(1)(1)
  13:  (1011) = (10)(1)(1)
  14:  (0111) = (0)(1)(1)(1)
  15:  (1111) = (1)(1)(1)(1)
  16: (00001) = (0)(0)(0)(0)(1)
  17: (10001) = (1000)(1)
  18: (01001) = (0)(100)(1)
  19: (11001) = (1100)(1)
  20: (00101) = (0)(0)(10)(1)

The non-"co" version is A211100.
Positions of 2's are A329357.
Numbers whose binary expansion is co-Lyndon are A275692.
Length of the co-Lyndon factorization of the binary expansion is A329312.
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A328596, A329131, A329314, A329318, A329324, A329325.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
colynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[colynfac[Drop[q,i]],Take[q,i]]]@Last[Select[Range[Length[q]],colynQ[Take[q,#]]&]]];
Table[Length[colynfac[Reverse[IntegerDigits[n,2]]]],{n,100}]

A329139 Numbers whose prime signature is an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their prime signatures begins:
   1: ()
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   7: (1)
   8: (3)
   9: (2)
  11: (1)
  12: (2,1)
  13: (1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  20: (2,1)
  23: (1)
  24: (3,1)
  25: (2)
  27: (3)

Complement of A329140.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is a necklace are A329138.
Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.

aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Last/@FactorInteger[#]]&]

A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167

Offset: 1

Gus Wiseman, May 05 2023

nonn

First differs from A329131 in lacking 450 and having 1500.

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

			The factorization of 90 is 2*3*3*5, modes {3}, so 90 is missing.
The factorization of 450 is 2*3*3*5*5, modes {3,5}, so 450 is missing.
The factorization of 900 is 2*2*3*3*5*5, modes {2,3,5}, so 900 is missing.
The factorization of 1500 is 2*2*3*5*5*5, modes {5}, so 1500 is present.
The terms together with their prime indices begin:
     2: {1}          27: {2,2,2}           67: {19}
     3: {2}          29: {10}              71: {20}
     4: {1,1}        31: {11}              73: {21}
     5: {3}          32: {1,1,1,1,1}       75: {2,3,3}
     7: {4}          37: {12}              79: {22}
     8: {1,1,1}      41: {13}              81: {2,2,2,2}
     9: {2,2}        43: {14}              83: {23}
    11: {5}          47: {15}              89: {24}
    13: {6}          49: {4,4}             97: {25}
    16: {1,1,1,1}    50: {1,3,3}           98: {1,4,4}
    17: {7}          53: {16}             101: {26}
    18: {1,2,2}      54: {1,2,2,2}        103: {27}
    19: {8}          59: {17}             107: {28}
    23: {9}          61: {18}             108: {1,1,2,2,2}
    25: {3,3}        64: {1,1,1,1,1,1}    109: {29}

First term with given bigomega is A000079.
For median instead of mode we have A053263.
Partitions of this type are counted by A362612.
A112798 lists prime indices, length A001222, sum A056239.
A356862 ranks partitions with a unique mode, counted by A362608.
A359178 ranks partitions with a unique co-mode, counted by A362610.
A362605 ranks partitions with more than one mode, counted by A362607.
A362606 ranks partitions with more than one co-mode, counted by A362609.
A362614 counts partitions by number of modes, ranked by A362611.
A362615 counts partitions by number of co-modes, ranked by A362613.
Cf. A000040, A002865, A327473, A327476, A358137, A360687.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[100],Commonest[prifacs[#]]=={Max[prifacs[#]]}&]

A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.

Original entry on oeis.org

1, 2, 3, 4, 7, 8, 10, 13, 15, 16, 22, 25, 31, 32, 36, 42, 46, 49, 53, 59, 63, 64, 76, 82, 94, 97, 109, 115, 127, 128, 136, 148, 156, 162, 166, 169, 170, 172, 181, 182, 190, 193, 201, 202, 211, 213, 214, 217, 221, 227, 235, 247, 255, 256, 280, 292, 306, 308

Offset: 1

Gus Wiseman, Nov 13 2019

nonn

We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
Conjecture: also numbers k such that the k-th composition in standard order (A066099) is a palindrome, cf. A025065, A242414, A317085, A317086, A317087, A335373. - Gus Wiseman, Jun 06 2020

			The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
   1:      () =               0 = 0
   2:     (0) =             (0) = (0)
   3:     (1) =             (1) = (1)
   4:    (00) =          (0)(0) = (0)(0)
   7:    (11) =          (1)(1) = (1)(1)
   8:   (000) =       (0)(0)(0) = (0)(0)(0)
  10:   (010) =         (0)(10) = (01)(0)
  13:   (101) =         (10)(1) = (1)(01)
  15:   (111) =       (1)(1)(1) = (1)(1)(1)
  16:  (0000) =    (0)(0)(0)(0) = (0)(0)(0)(0)
  22:  (0110) =        (0)(110) = (011)(0)
  25:  (1001) =        (100)(1) = (1)(001)
  31:  (1111) =    (1)(1)(1)(1) = (1)(1)(1)(1)
  32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
  36: (00100) =     (0)(0)(100) = (001)(0)(0)
  42: (01010) =     (0)(10)(10) = (01)(01)(0)
  46: (01110) =       (0)(1110) = (0111)(0)
  49: (10001) =       (1000)(1) = (1)(0001)
  53: (10101) =     (10)(10)(1) = (1)(01)(01)
  59: (11011) =     (110)(1)(1) = (1)(1)(011)
  63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)

Lyndon and co-Lyndon compositions are (both) counted by A059966.
Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
Cf. A329358, A329399, A329400, A329401.

lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Select[Range[100],Length[lynfac[Rest[IntegerDigits[#,2]]]]==Length[colynfac[Rest[IntegerDigits[#,2]]]]&]

A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 1, 4, 2, 5, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 3, 3, 1, 4, 1, 1, 1, 4, 1, 2, 4, 2, 1, 4, 3, 5, 1, 1, 5, 2, 6, 1, 7

Offset: 1

Gus Wiseman, Jan 12 2017

The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details.

These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019

			The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are:
             (1) = (1)
            (12) = (1)*(2)
           (122) = (1)*(2)*(2)
          (1221) = (1)*(221)
         (12211) = (1)*(2211)
        (122112) = (1)*(2)*(2211)
       (1221121) = (1)*(221121)
      (12211212) = (1)*(2)*(221121)
     (122112122) = (1)*(2)*(2)*(221121)
    (1221121221) = (1)*(221)*(221121)
   (12211212212) = (1)*(2)*(221)*(221121)
  (122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1    2    2 1    2 1 1    2 1 1 1    2 1 1 1 1    2 1 1 1 1 1
          3      3 1      2 2 1      2 2 1 1      2 1 2 1 1
                 4        3 1 1      3 1 1 1      2 2 1 1 1
                          3 2        3 1 2        2 2 2 1
                          4 1        3 2 1        3 1 1 1 1
                          5          4 1 1        3 1 1 2
                                     4 2          3 1 2 1
                                     5 1          3 2 1 1
                                     6            3 2 2
                                                  3 3 1
                                                  4 1 1 1
                                                  4 1 2
                                                  4 2 1
                                                  4 3
                                                  5 1 1
                                                  5 2
                                                  6 1
                                                  7.

Cf. A000740, A215474, A228369, A277427.
The binary version is A329318.
The binary non-"co" version is A102659.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is co-Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Normal Lyndon words are A060223.
Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],colynQ],lexsort],{n,5}] (* Gus Wiseman, Nov 15 2019 *)

Row lengths are A059966(n) = number of prime compositions of n.

A329138 Numbers whose prime signature is a necklace.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

First differs from A304678 in having 1350 = 2^1 * 3^3 * 5^2. First differs from A316529 in having 150 = 2^1 * 3^1 * 5^2.
A number's prime signature (A124010) is the sequence of positive exponents in its prime factorization.
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.

			The sequence of terms together with their prime signatures begins:
   2: (1)
   3: (1)
   4: (2)
   5: (1)
   6: (1,1)
   7: (1)
   8: (3)
   9: (2)
  10: (1,1)
  11: (1)
  13: (1)
  14: (1,1)
  15: (1,1)
  16: (4)
  17: (1)
  18: (1,2)
  19: (1)
  21: (1,1)
  22: (1,1)

Complement of A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Numbers whose reversed binary expansion is a necklace are A328595.
Numbers whose prime signature is a Lyndon word are A329131.
Numbers whose prime signature is aperiodic are A329139.
Cf. A001037, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329140.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Select[Range[2,100],neckQ[Last/@FactorInteger[#]]&]

cp's OEIS Frontend

A102659 List of Lyndon words on {1,2} sorted first by length and then lexicographically.

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A211100 Number of factors in Lyndon factorization of binary expansion of n.

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A329312 Length of the co-Lyndon factorization of the binary expansion of n.

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A329313 Length of the Lyndon factorization of the reversed binary expansion of n.

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A329326 Length of the co-Lyndon factorization of the reversed binary expansion of n.

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A329139 Numbers whose prime signature is an aperiodic word.

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A362616 Numbers in whose prime factorization the greatest factor is the unique mode.

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A329395 Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.

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A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.