A027375 -id:A027375 - cp's OEIS frontend

A323867 Number of aperiodic arrays of positive integers summing to n.

1, 1, 1, 5, 11, 33, 57, 157, 303, 683, 1358, 2974, 5932, 12560, 25328, 52400, 106256, 217875, 441278, 899955, 1822703, 3701401, 7491173, 15178253, 30691135, 62085846, 125435689, 253414326, 511547323, 1032427635, 2082551931, 4199956099, 8466869525, 17064777665

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

The 1-dimensional case is A000740.

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(5) = 33 arrays:
  5  14  23  32  41  113  122  131  212  221  311  1112  1121  1211  2111
.
  1  2  3  4  11  11  12  21
  4  3  2  1  12  21  11  11
.
  1  1  1  2  2  3
  1  2  3  1  2  1
  3  2  1  2  1  1
.
  1  1  1  2
  1  1  2  1
  1  2  1  1
  2  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000670, A000740, A027375, A101509.

Cf. A323860, A323861, A323862, A323863, A323864, A323866, A323869.

List([0..30], A323867); # See A323861 for code; Andrew Howroyd, Aug 21 2019

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnmats[n_]:=Union@@Permutations/@Select[Union@@(Tuples[Permutations/@#]&/@Map[primeMS,facs[n],{2}]),SameQ@@Length/@#&];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Union@@Table[Select[ptnmats[k],apermatQ],{k,Times@@Prime/@#&/@IntegerPartitions[n]}]],{n,15}]

Terms a(16) and beyond from Andrew Howroyd, Aug 21 2019

A324461 Number of simple graphs with n vertices and distinct rotations.

Original entry on oeis.org

1, 1, 0, 6, 48, 1020, 32232, 2097144, 268369920, 68719472640, 35184338533920, 36028797018963936, 73786976226114539520, 302231454903657293676480, 2475880078570197599844819072, 40564819207303340847860140736640, 1329227995784915854457062986570792960

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 48 graphs with distinct rotations.

Cf. A000088, A000740, A003436, A006125, A027375, A192314, A192332, A306669, A306715, A323860, A323864, A323867, A324462 (covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,0,5}]

a(n)={if(n==0, 1, sumdiv(n, d, moebius(d)*2^(n*(n/d-1)/2 + n*(d\2)/d)))} \\ Andrew Howroyd, Aug 15 2019

from sympy import mobius, divisors
def A324461(n): return sum(mobius(m:=n//d)<<(n*(d-1)>>1)+d*(m>>1) for d in divisors(n,generator=True)) if n else 1 # Chai Wah Wu, Jul 03 2024

a(n > 0) = A306715(n) * n.

a(n) = Sum_{d|n} mu(d)*2^(n*(n/d-1)/2 + n*floor(d/2)/d) for n > 0. - Andrew Howroyd, Aug 15 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 15 2019

A045655 Number of 2n-bead balanced binary strings, rotationally equivalent to reversed complement.

Original entry on oeis.org

1, 2, 6, 20, 54, 152, 348, 884, 1974, 4556, 10056, 22508, 48636, 106472, 228444, 491120, 1046454, 2228192, 4713252, 9961436, 20960904, 44038280, 92252100, 192937940, 402599676, 838860152, 1744723896, 3623869388, 7515962172

Offset: 0

David W. Wilson

nonn

a(n) is the number of ordered pairs (a,b) of length n binary sequences such that a and b are equivalent by rotational symmetry. - Geoffrey Critzer, Dec 31 2011
a(n) is the weighted sum of binary strings of length n by their number of distinct images by rotation. There is a natural correspondence between the first 2^(n-1) sequences (starting with a 0) and the 2^(n-1) starting with a 1 by inversion. There is also an internal correspondance by order inversion. - Olivier Gérard, Jan 01 2011
The number of k-circulant n X n (0,1) matrices, which means the number of n X n binary matrices where rows from the 2nd row on are obtained from the preceding row by a cyclic shift by k columns for some 0 <= k < n.  - R. J. Mathar, Mar 11 2017

			a(2)= 6 because there are 6 such ordered pairs of length 2 binary sequences: (00,00),(11,11),(01,01),(10,10),(01,10),(10,01).
a(3)= 20 because the classes of 3-bit strings are 1*(000), 3*(001,010,100), 3*(011,110,101), 1*(111) = 1 + 9 + 9 + 1.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Chuan Guo, J. Shallit, A. M. Shur, On the Combinatorics of Palindromes and Antipalindromes, arXiv preprint arXiv:1503.09112 [cs.FL], 2015.
V. V. Strok, Circulant matrices and the spectra of de Bruijn graphs, Ukr. Math. J. 44 (11) (1992) 1446-1454.

Cf. A000031 counts the string classes.

Cf. A000984, A023900, A045664, A045653, A045654, A045656.

f[n_] := 2*Plus @@ Table[ Length[ Union[ NestList[ RotateLeft, IntegerDigits[b, 2, n], n - 1]]], {b, 0, 2^(n - 1) - 1}]; f[0] = 1; Array[f, 21, 0] (* Olivier Gérard, Jan 01 2012 *)

c(n)={sumdiv(n,d, moebius(d)*d)} \\ A023900
a(n)={if(n<1, n==0, sumdiv(n, d, c(n/d)*d*2^d))} \\ Andrew Howroyd, Sep 15 2019

For n >= 1, a(n) = Sum_{d|n} A045664(d) = Sum_{d|n} d*A027375(d) = Sum_{d|n} d^2*A001037(d).

a(n) = Sum_{d|n} A023900(n/d)*d*2^d. - Andrew Howroyd, Sep 15 2019

A329317 Length of the Lyndon factorization of the reversed first n terms of A000002.

Original entry on oeis.org

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

Offset: 1

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. 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).

			The sequence of Lyndon factorizations of the reversed initial terms of A000002 begins:
   1: (1)
   2: (2)(1)
   3: (2)(2)(1)
   4: (122)(1)
   5: (1122)(1)
   6: (2)(1122)(1)
   7: (12)(1122)(1)
   8: (2)(12)(1122)(1)
   9: (2)(2)(12)(1122)(1)
  10: (122)(12)(1122)(1)
  11: (2)(122)(12)(1122)(1)
  12: (2)(2)(122)(12)(1122)(1)
  13: (122)(122)(12)(1122)(1)
  14: (112212212)(1122)(1)
  15: (2)(112212212)(1122)(1)
  16: (12)(112212212)(1122)(1)
  17: (1121122122121122)(1)
  18: (2)(1121122122121122)(1)
  19: (2)(2)(1121122122121122)(1)
  20: (122)(1121122122121122)(1)
For example, the reversed first 13 terms of A000002 are (1221221211221), with Lyndon factorization (122)(122)(12)(1122)(1), so a(13) = 5.

Row-lengths of A329316.
The non-reversed version is A329315.
Cf. A000002, A000031, A001037, A027375, A059966, A060223, A088568, A102659, A211100, A288605, A296372, A296658, A329314, 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,#]]&]]]];
kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Table[Length[lynfac[Reverse[kol[n]]]],{n,100}]

A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

Original entry on oeis.org

1, 4, 24, 212, 1080, 18672, 94584, 2182752, 21261708, 408988080, 3245265144, 168549358368, 1053716696760, 42565371692592, 921132763909200, 26578273403903040, 260741534058271800, 20313207979498492344, 185603174638656822264, 16066126777465282744800, 324499299994016295338064

Offset: 1

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(3) = 24 matrices:
  [123][132][213][312][231][321][122][211][112][221][121][212]
.
  [1][1][2][3][2][3][1][2][1][2][1][2]
  [2][3][1][1][3][2][2][1][1][2][2][1]
  [3][2][3][2][1][1][2][1][2][1][1][2]

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

Cf. A000670, A000740, A027375, A060223, A265627.

Cf. A323860, A323864, A323867, A323868, A323871.

List([1..30], A323869); # See A323861 for code; Andrew Howroyd, Aug 21 2019

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[nrmmats[n],apermatQ]],{n,6}]

a(n) = n*A323871(n). - Andrew Howroyd, Aug 21 2019

Terms a(9) and beyond from Andrew Howroyd, Aug 21 2019

A329316 Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the reversed first n terms of A000002.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 4, 1, 1, 4, 1, 2, 4, 1, 1, 2, 4, 1, 1, 1, 2, 4, 1, 3, 2, 4, 1, 1, 3, 2, 4, 1, 1, 1, 3, 2, 4, 1, 3, 3, 2, 4, 1, 9, 4, 1, 1, 9, 4, 1, 2, 9, 4, 1, 16, 1, 1, 16, 1, 1, 1, 16, 1, 3, 16, 1, 1, 3, 16, 1, 5, 16, 1, 6, 16, 1, 1, 6, 16, 1, 2, 6

Offset: 0

Gus Wiseman, Nov 11 2019

There are no repeated rows, as row n has sum n.
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).
It appears that some numbers (such as 10) never appear in the sequence.

			Triangle begins:
   1: (1)
   2: (1,1)
   3: (1,1,1)
   4: (3,1)
   5: (4,1)
   6: (1,4,1)
   7: (2,4,1)
   8: (1,2,4,1)
   9: (1,1,2,4,1)
  10: (3,2,4,1)
  11: (1,3,2,4,1)
  12: (1,1,3,2,4,1)
  13: (3,3,2,4,1)
  14: (9,4,1)
  15: (1,9,4,1)
  16: (2,9,4,1)
  17: (16,1)
  18: (1,16,1)
  19: (1,1,16,1)
  20: (3,16,1)
For example, the reversed first 13 terms of A000002 are (1221221211221), with Lyndon factorization (122)(122)(12)(1122)(1), so row 13 is (3,3,2,4,1).

Row lengths are A329317.
The non-reversed version is A329315.
Cf. A000002, A000031, A001037, A027375, A059966, A060223, A088568, A102659, A211100, A288605, A296372, A296658, A329314, 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,#]]&]]]];
kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],q[[-2]],Last[q]},{1,1,1},0,{1,1,2},1,{1,2,1},2,{1,2,2},0,{2,1,1},2,{2,1,2},2,{2,2,1},1,{2,2,2},1]]]
kol[n_Integer]:=Nest[kolagrow,{1},n-1];
Table[Length/@lynfac[Reverse[kol[n]]],{n,100}]

A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

Original entry on oeis.org

0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 39, 41, 43, 47, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 135, 137, 138, 139, 141, 143, 145, 146, 147, 149, 151, 155, 159, 161, 163

Offset: 1

Gus Wiseman, Apr 22 2020

nonn

Reversed Lyndon words are different from co-Lyndon words (A326774).
A Lyndon word is a finite sequence of positive integers that is lexicographically strictly less than all of its cyclic rotations.
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 sequence of all reversed Lyndon words begins:
    0: ()            37: (3,2,1)         83: (2,3,1,1)
    1: (1)           39: (3,1,1,1)       85: (2,2,2,1)
    2: (2)           41: (2,3,1)         87: (2,2,1,1,1)
    4: (3)           43: (2,2,1,1)       91: (2,1,2,1,1)
    5: (2,1)         47: (2,1,1,1,1)     95: (2,1,1,1,1,1)
    8: (4)           64: (7)            128: (8)
    9: (3,1)         65: (6,1)          129: (7,1)
   11: (2,1,1)       66: (5,2)          130: (6,2)
   16: (5)           67: (5,1,1)        131: (6,1,1)
   17: (4,1)         68: (4,3)          132: (5,3)
   18: (3,2)         69: (4,2,1)        133: (5,2,1)
   19: (3,1,1)       71: (4,1,1,1)      135: (5,1,1,1)
   21: (2,2,1)       73: (3,3,1)        137: (4,3,1)
   23: (2,1,1,1)     74: (3,2,2)        138: (4,2,2)
   32: (6)           75: (3,2,1,1)      139: (4,2,1,1)
   33: (5,1)         77: (3,1,2,1)      141: (4,1,2,1)
   34: (4,2)         79: (3,1,1,1,1)    143: (4,1,1,1,1)
   35: (4,1,1)       81: (2,4,1)        145: (3,4,1)

The non-reversed version is A275692.
The generalization to necklaces is A333943.
The dual version (reversed co-Lyndon words) is A328596.
The case that is also co-Lyndon is A334266.
Binary Lyndon words are counted by A001037.
Lyndon compositions are counted by A059966.
Normal Lyndon words are counted by A060223.
Numbers whose prime signature is a reversed Lyndon word are A334298.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Reversed Lyndon words are A334265 (this sequence).
- Co-Lyndon words are A326774.
- Reversed co-Lyndon words are A328596.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Lyndon factorizations are counted by A333940.
- Length of Lyndon factorization of reverse is A334297.
Cf. A000031, A027375, A138904, A211100, A328595, A329131, A329313, A333764, A334028, A334267.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
lynQ[q_]:=Length[q]==0||Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,Length[q]-1,1,And];
Select[Range[0,100],lynQ[Reverse[stc[#]]]&]

A054718 Number of ternary sequences with primitive period n.

Original entry on oeis.org

1, 3, 6, 24, 72, 240, 696, 2184, 6480, 19656, 58800, 177144, 530640, 1594320, 4780776, 14348640, 43040160, 129140160, 387400104, 1162261464, 3486725280, 10460350992, 31380882456, 94143178824, 282428998560, 847288609200, 2541864234000, 7625597465304

Offset: 0

N. J. A. Sloane, Apr 20 2000

nonn

Equivalently, output sequences with primitive period n from a simple cycling shift register.

Seiichi Manyama, Table of n, a(n) for n = 0..2095 (terms 0..650 from Alois P. Heinz)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column k=3 of A143324.

Cf. A027375, A054719, A054720, A054721.

with(numtheory):
a:= n-> `if`(n=0, 1, add(mobius(d)*3^(n/d), d=divisors(n))):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 21 2012

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*3^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

a(n) = if(n==0,1,sumdiv(n,d, moebius(d) * 3^(n/d) )); \\ Joerg Arndt, Apr 14 2013

a(n) = Sum_{d|n} mu(d)*3^(n/d).
a(0) = 1, a(n) = n * A027376(n).
a(n) = 3 * A034741(n).
G.f.: 1 + 3 * Sum_{k>=1} mu(k) * x^k / (1 - 3*x^k). - Ilya Gutkovskiy, Apr 14 2021

A265627 Number of n X n "primitive" binary matrices.

Original entry on oeis.org

2, 10, 498, 65040, 33554370, 68718945018, 562949953421058, 18446744065119682560, 2417851639229258080977408, 1267650600228227149696920981450, 2658455991569831745807614120560685058, 22300745198530623141526273539119741048774160

Offset: 1

Jeffrey Shallit, Dec 10 2015

nonn

A rectangular matrix is "primitive" in this sense if it cannot be expressed as a "tiling" of a single smaller matrix repeated in both directions.
Thus, for example, the 2 X 2 matrix with both rows equal to [1,0] is not primitive, since it can "tiled" by a single row.
This is the 2-dimensional generalization of A027375.

			We see a(2) = 10 since there are 16 possible 2 X 2 binary matrices, two are excluded because all their entries are the same, and four more are excluded because they are [[1,0],[1,0]] or a transpose or a negation.

Cf. A027375.

with(numtheory):
prim := proc(k,m,n) option remember;
        dm := divisors(m);
        dn := divisors(n);
        s := 0;
        for d1 in dm do
                for d2 in dn do
                        s := s+(k^(m*n/(d1*d2)))*mobius(d1)*mobius(d2);
                        od;
                od;
        s;
        end:
seq(prim(2,n,n), n=1..40);

prim[k_, m_, n_] := prim[k, m, n] = Module[{s = 0},
Do[Do[s = s + (k^(m*n/(d1*d2)))*MoebiusMu[d1]*MoebiusMu[d2],
   {d1, Divisors[m]}], {d2, Divisors[n]}]; s];
a[n_] := prim[2, n, n]
Table[a[n], {n, 1, 12}] (* Jean-François Alcover, Jul 24 2022, after Maple code *)

A general formula for the number of m X n "primitive" matrices over an alphabet of size k is Sum_{d|m, e|n} k^{m*n/(d*e)}*mu(d)*mu(e), where mu is the Möbius function.

A296975 Number of aperiodic normal sequences of length n.

Original entry on oeis.org

1, 2, 12, 72, 540, 4668, 47292, 545760, 7087248, 102247020, 1622632572, 28091562840, 526858348380, 10641342923148, 230283190977300, 5315654681435520, 130370767029135900, 3385534663249753392, 92801587319328411132, 2677687796244281955480, 81124824998504073834516

Offset: 1

Gus Wiseman, Dec 22 2017

nonn

A finite sequence is normal if it spans an initial interval of positive integers. It is aperiodic if every cyclic rotation is different.

			The a(3) = 12 aperiodic normal sequences are 112, 121, 122, 123, 132, 211, 212, 213, 221, 231, 312, 321.
The 15 non-aperiodic normal sequences of length 6 are: 111111, 112112, 121121, 121212, 122122, 123123, 132132, 211211, 212121, 212212, 213213, 221221, 231231, 312312, 321321.

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

Cf. A000670, A000740, A001037, A019536, A027375, A060223, A095684, A185700, A296976, A296977, A296978.

Table[DivisorSum[n,MoebiusMu[n/#]*Sum[k!*StirlingS2[#,k],{k,#}]&],{n,25}]

\\ here b(n) is A000670.
b(n)={polcoef(serlaplace(1/(2-exp(x+O(x*x^n)))),n)}
a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ Andrew Howroyd, Aug 29 2018

a(n) = n * A060223(n) = Sum_{d|n} mu(d) * A000670(n/d).

cp's OEIS Frontend

A323867 Number of aperiodic arrays of positive integers summing to n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

Extensions

A324461 Number of simple graphs with n vertices and distinct rotations.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A045655 Number of 2n-bead balanced binary strings, rotationally equivalent to reversed complement.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A329317 Length of the Lyndon factorization of the reversed first n terms of A000002.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A323869 Number of aperiodic matrices of size n whose entries cover an initial interval of positive integers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Mathematica

Formula

Extensions

A329316 Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the reversed first n terms of A000002.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A334265 Numbers k such that the k-th composition in standard order is a reversed Lyndon word.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A054718 Number of ternary sequences with primitive period n.

Views