A318731 -id:A318731 - cp's OEIS frontend

A329131 Numbers whose prime signature is a Lyndon word.

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, Nov 06 2019

nonn

First differs from A133811 in having 50.
A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.
A number's prime signature is the sequence of positive exponents in its prime factorization.

			The prime signature of 30870 is (1,2,1,3), which is a Lyndon word, so 30870 is in the sequence.
The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   18: {1,2,2}
   19: {8}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Numbers whose reversed binary expansion is Lyndon are A328596.
Numbers whose prime signature is a necklace are A329138.
Numbers whose prime signature is aperiodic are A329139.
Lyndon compositions are A059966.
Prime signature is A124010.
Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.

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

Intersection of A329138 and A329139.

A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304

Offset: 1

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.
A periodic palindrome is just a necklace that is equivalent to its reverse. The number of binary periodic palindromes of length n is given by A164090(n). A binary periodic palindrome can only be equivalent to its complement when there are an equal number of 0's and 1's. - Andrew Howroyd, Sep 29 2017
Number of cyclic compositions (necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic cyclic compositions begins:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (113)    (33)      (115)
                    (112)   (122)    (114)     (133)
                    (1111)  (11111)  (222)     (223)
                                     (1122)    (11113)
                                     (11112)   (11212)
                                     (111111)  (11122)
                                               (1111111)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A179181.
Cf. A016116, A045674, A056508, A164090, A285012.
Cf. A000740, A000837, A008965, A025065, A059966, A242414, A296302, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[q,And[Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And],Array[SameQ[RotateRight[q,#],Reverse[RotateRight[q,#]]]&,Length[q],1,Or]]]]],{n,15}] (* Gus Wiseman, Sep 16 2018 *)

a(2n+1) = A164090(2n+1)/2 = 2^n, a(2n) = (A164090(2n) + A045674(n))/2. - Andrew Howroyd, Sep 29 2017

a(17)-a(45) from Andrew Howroyd, Apr 07 2017

A318745 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 19, 32, 53, 94, 158, 279, 480, 847, 1487, 2647, 4676, 8349, 14865, 26630, 47700, 85778, 154290, 278318, 502437, 908880, 1645713, 2984546, 5417743, 9847189, 17914494, 32625523, 59467893, 108493134, 198089610, 361965238, 661883231, 1211161991

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(7) = 12 Lyndon compositions with adjacent parts coprime:
  (7)
  (16) (25) (34)
  (115)
  (1114) (1213) (1132) (1123)
  (11113) (11212)
  (111112)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A100953, A167606, A296302, A318728, A318731, A318746, A318747.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, (n > 1) + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

a(n) = A328669(n) + 1 for n > 1. - Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A329141 Number of Lyndon compositions of n that are not weakly increasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 11, 28, 60, 131, 263, 530, 1029, 2009, 3853, 7414, 14152, 27105, 51755, 99069, 189558, 363468, 697302, 1340220, 2578362, 4968001, 9582682, 18508226, 35784670, 69266825, 134207336, 260290846, 505274108, 981691926

Offset: 1

Gus Wiseman, Nov 10 2019

nonn

A Lyndon composition of n is a finite sequence of positive integers summing to n that is lexicographically strictly less than all of its cyclic rotations.

			The a(6) = 1 through a(8) = 11 compositions:
  (132)  (142)    (143)
         (1132)   (152)
         (1213)   (1142)
         (11212)  (1214)
                  (1232)
                  (1322)
                  (11132)
                  (11213)
                  (11312)
                  (12122)
                  (111212)

Lyndon compositions are A059966.
Lyndon compositions that are weakly increasing are A167934.
Binary Lyndon words are A001037.
Necklace compositions are A008965.
Cf. A000031, A000740, A178472, A318731, A328596, A329131, A329145.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!OrderedQ[#]&&neckQ[#]&&aperQ[#]&]],{n,10}]

a(n) = A059966(n) - A167934(n).

A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 7, 10, 14, 21, 31, 42, 63, 91, 123, 184, 255, 371, 511, 750, 1015, 1519, 2047, 3030, 4092, 6111, 8176, 12222, 16383, 24486, 32767, 49024, 65503, 98175, 131061, 196308, 262143, 392959, 524223, 785910, 1048575, 1572256, 2097151, 3144702, 4194162

Offset: 0

Marks R. Nester

nonn

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure.

Number of Lyndon compositions (aperiodic necklaces of positive integers) summing to n that can be rotated to form a palindrome. - Gus Wiseman, Sep 16 2018

			From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic Lyndon compositions begins:
  (1)  (2)  (3)  (4)    (5)    (6)      (7)
                 (112)  (113)  (114)    (115)
                        (122)  (1122)   (133)
                               (11112)  (223)
                                        (11113)
                                        (11212)
                                        (11122)
(End)

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Row sums of A179317.

Cf. A008965, A025065, A056476, A059966, A242414, A285037, A317085, A317086, A317087, A318731.

(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a56503[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])];
a[n_] := DivisorSum[n, MoebiusMu[#] a56503[n/#]&];
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)

a(n) = {if(n < 1, n==0, sumdiv(n, d, moebius(d)*(2 + d%2)*(2^(n/d\2)))/(4 - n%2))} \\ Andrew Howroyd, Sep 26 2019

seq(n) = Vec(1 + (1/2)*sum(k=1, n, moebius(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - moebius(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)) + O(x*x^n))) \\ Andrew Howroyd, Sep 27 2019

a(n) = Sum_{d|n} mu(d)*A056503(n/d) for n > 0.
a(n) = Sum_{k=1..2} A285037(n, k). - Andrew Howroyd, Apr 08 2017
G.f.: 1 + (1/2)*Sum_{k>=1} mu(k)*x^k*(2 + 3*x^k)/(1 - 2*x^(2*k)) - mu(2*k)*x^(2*k)*(1 + x^(2*k))/(1 - 2*x^(4*k)). - Andrew Howroyd, Sep 27 2019

a(17)-a(45) from Andrew Howroyd, Apr 08 2017

a(0)=1 prepended by Andrew Howroyd, Sep 27 2019

A318746 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 4, 5, 6, 8, 11, 17, 20, 29, 41, 56, 79, 107, 155, 214, 305, 422, 604, 850, 1207, 1709, 2424, 3439, 4905, 6972, 9949, 14171, 20268, 28915, 41392, 59176, 84790, 121428, 174163, 249760, 358578, 514873, 739910, 1063523, 1529767, 2200926

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 17 Lyndon compositions with successive parts indivisible:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,3,9) (2,5,7) (2,7,5) (3,4,7) (3,6,5) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,4,3,5) (2,4,5,3) (2,5,4,3)
  (2,3,2,4,3)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318747.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,LyndonQ[#]&&And@@Not/@Divisible@@@Partition[#,2,1,1]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->i%j<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 2, 3, 5, 5, 8, 7, 12, 14, 20, 31, 37, 51, 64, 96, 129, 177, 246, 328, 465, 630, 889, 1230, 1692, 2370, 3250, 4587, 6354, 8895, 12384, 17252, 24180, 33777, 47336, 66254, 92752, 130142, 182337, 256246, 359500, 505231, 709787, 997951, 1403883

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

			The a(14) = 12 Lyndon compositions with adjacent parts indivisible either way:
  (14)
  (3,11) (4,10) (5,9) (6,8)
  (2,5,7) (2,7,5) (3,4,7) (3,7,4)
  (2,3,2,7) (2,3,4,5) (2,5,4,3)

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A000740, A008965, A059966, A285573, A303362, A318726, A318727, A318729, A318730, A318731, A318745, A318746.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[Length[#]==1,And[LyndonQ[#],And@@Not/@Divisible@@@Partition[#,2,1,1],And@@Not/@Divisible@@@Reverse/@Partition[#,2,1,1]]]&]],{n,20}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->i%j<>0 && j%i<>0))); vector(n, n, 1 + sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Sep 08 2018

A328669 Number of Lyndon compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.

Original entry on oeis.org

1, 0, 1, 2, 4, 6, 11, 18, 31, 52, 93, 157, 278, 479, 846, 1486, 2646, 4675, 8348, 14864, 26629, 47699, 85777, 154289, 278317, 502436, 908879, 1645712, 2984545, 5417742, 9847188, 17914493, 32625522, 59467892, 108493133, 198089609, 361965237, 661883230, 1211161990

Offset: 1

Gus Wiseman, Oct 26 2019

nonn

A Lyndon composition of n is a finite sequence of positive integers summing to n that is lexicographically strictly less than all of its cyclic rotations.

			The a(1) = 1 through a(8) = 18 Lyndon compositions (empty column not shown):
  (1)  (12)  (13)   (14)    (15)     (16)      (17)
             (112)  (23)    (114)    (25)      (35)
                    (113)   (123)    (34)      (116)
                    (1112)  (132)    (115)     (125)
                            (1113)   (1114)    (134)
                            (11112)  (1123)    (143)
                                     (1132)    (152)
                                     (1213)    (1115)
                                     (11113)   (1214)
                                     (11212)   (1232)
                                     (111112)  (11114)
                                               (11123)
                                               (11132)
                                               (11213)
                                               (11312)
                                               (111113)
                                               (111212)
                                               (1111112)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The non-Lyndon version is A328609 or A318748 (with singletons).
The non-Lyndon non-circular version is A167606.
The version with singletons is A318745.
The necklace case is A328597 or A318728 (with singletons).
The aperiodic case is A328670.
Lyndon compositions are A059966, with relatively prime case A318731.
Cf. A000031, A000740, A008965, A328172, A328601, A328602.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],aperQ[#]&&neckQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, k*b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, sumdiv(n, d, moebius(d)*v[n/d])/n)} \\ Andrew Howroyd, Nov 01 2019

a(n > 1) = A318745(n) - 1.

A328670 Number of aperiodic compositions of n where every pair of adjacent parts (including the last with the first) is relatively prime.

Original entry on oeis.org

1, 0, 2, 5, 11, 20, 41, 75, 147, 272, 533, 976, 1881, 3490, 6616, 12378, 23405, 43781, 82536, 154709, 291043, 546139, 1026685, 1927038, 3621004, 6798417, 12770935, 23980791, 45042957, 84584416, 158863805, 298336153, 560302805, 1052234995, 1976157456, 3711209272

Offset: 1

Gus Wiseman, Oct 29 2019

nonn

A sequence is aperiodic if all of its cyclic rotations are different.

			The a(1) = 1 through a(6) = 20 compositions (empty column not shown):
  (1)  (12)  (13)   (14)    (15)
       (21)  (31)   (23)    (51)
             (112)  (32)    (114)
             (121)  (41)    (123)
             (211)  (113)   (132)
                    (131)   (141)
                    (311)   (213)
                    (1112)  (231)
                    (1121)  (312)
                    (1211)  (321)
                    (2111)  (411)
                            (1113)
                            (1131)
                            (1311)
                            (3111)
                            (11112)
                            (11121)
                            (11211)
                            (12111)
                            (21111)

Andrew Howroyd, Table of n, a(n) for n = 1..200

The non-aperiodic version is A328609 or A318748 (with singletons).
The non-aperiodic, non-circular version is A167606.
The Lyndon word case is A328669.
Lyndon compositions are A059966, with relatively prime case A318731.
Cf. A000031, A000740, A008965, A318728, A328172, A328597, A328601, A328602.

aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],aperQ[#]&&And@@CoprimeQ@@@Partition[#,2,1,1]&]],{n,10}]

b(n, q, pred)={my(M=matrix(n, n)); for(k=1, n, M[k, k]=pred(q, k); for(i=1, k-1, M[i, k]=sum(j=1, k-i, if(pred(j, i), M[j, k-i], 0)))); M[q, ]}
seq(n)={my(v=sum(k=1, n, b(n, k, (i, j)->gcd(i, j)==1))); vector(n, n, sumdiv(n, d, moebius(d)*v[n/d]))} \\ Andrew Howroyd, Nov 01 2019

Terms a(21) and beyond from Andrew Howroyd, Nov 01 2019

A329145 Number of non-necklace compositions of n.

Original entry on oeis.org

0, 0, 1, 3, 9, 19, 45, 93, 197, 405, 837, 1697, 3465, 7011, 14193, 28653, 57825, 116471, 234549, 471801, 948697, 1906407, 3829581, 7689357, 15435033, 30973005, 62137797, 124630149, 249922665, 501078345, 1004468157, 2013263853, 4034666121, 8084640465

Offset: 1

Gus Wiseman, Nov 10 2019

nonn

A necklace composition of n is a finite sequence of positive integers summing to n that is lexicographically minimal among all of its cyclic rotations.

			The a(3) = 1 through a(6) = 19 compositions:
  (21)  (31)   (32)    (42)
        (121)  (41)    (51)
        (211)  (131)   (141)
               (212)   (213)
               (221)   (231)
               (311)   (312)
               (1121)  (321)
               (1211)  (411)
               (2111)  (1131)
                       (1221)
                       (1311)
                       (2112)
                       (2121)
                       (2211)
                       (3111)
                       (11121)
                       (11211)
                       (12111)
                       (21111)

Numbers whose prime signature is not a necklace are A329142.
Binary necklaces are A000031.
Necklace compositions are A008965.
Lyndon compositions are A059966.
Numbers whose reversed binary expansion is a necklace are A328595.
Cf. A000740, A001037, A019536, A070211, A178472, A318731, A329138, A329141.

neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!neckQ[#]&]],{n,10}]

a(n) = 2^(n-1) - A008965(n).

cp's OEIS Frontend

A329131 Numbers whose prime signature is a Lyndon word.

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A056503 Number of periodic palindromic structures of length n using a maximum of two different symbols.

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A318745 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being coprime.

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A329141 Number of Lyndon compositions of n that are not weakly increasing.

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A056513 Number of primitive (period n) periodic palindromic structures using a maximum of two different symbols.

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A318746 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and successive parts (including the last with the first part) being indivisible.

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A318747 Number of Lyndon compositions (aperiodic necklaces of positive integers) with sum n and adjacent parts (including the last with the first part) being indivisible (either way).

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