A027375 -id:A027375 - cp's OEIS frontend

A138904 Number of rotational symmetries in the binary expansion of a number.

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

Offset: 0

Max Sills, Apr 03 2008, Apr 04 2008

Mersenne numbers of form (2^n - 1) have n rotational symmetries.
For prime length binary expansions these are the only nontrivial symmetries.
For composite length expansions it seems that when the number of symmetries is nontrivial it is equal to a factor of the length. We're working on an explicit formula.
Discovered in the context of random circulant matrices, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.
When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1 and value b = 0.
Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.
Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.
Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).
Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetries. That number is 2^n - 1. For each proper divisor d (excluding 1), you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.
Also the number of rotational symmetries of the n-th composition in standard order (graded reverse-lexicographic). This composition (row n of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. - Gus Wiseman, Apr 19 2020
From Gus Wiseman, Apr 19 2020: (Start)
Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Period of binary expansion is A302291.
Compositions by sum and number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Reversed co-necklaces are A328595.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.
(End).

			a(10) = 2 because the binary expansion of 10 is 1010 and it has two rotational symmetries (including identity).

Maxwell Sills and Gary Doran, Table of n, a(n) for n = 0..99

Cf. A136441, A138954.

Table[IntegerLength[n,2]/Length[Union[Array[RotateRight[IntegerDigits[n,2],#]&,IntegerLength[n,2]]]],{n,100}] (* Gus Wiseman, Apr 19 2020 *)

a(n) = A070939(n)/A302291(n) = A000120(n)/A333632(n). - Gus Wiseman, Apr 19 2020

A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 4, 4, 0, 0, 5, 10, 10, 5, 0, 0, 6, 12, 18, 12, 6, 0, 0, 7, 21, 35, 35, 21, 7, 0, 0, 8, 24, 56, 64, 56, 24, 8, 0, 0, 9, 36, 81, 126, 126, 81, 36, 9, 0, 0, 10, 40, 120, 200, 250, 200, 120, 40, 10, 0, 0, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Clark Kimberling

			For example, T(4,2) counts 1100,1001,0011,0110; T(2,1) counts 10, 01 (hence also counts 1010, 0101).
Rows:
  1;
  1,  1;
  0,  2,  0;
  0,  3,  3,  0;
  0,  4,  4,  4,  0;
  0,  5, 10, 10,  5,  0;

M. F. Hasler, A050186, rows 0..50, Sep 27 2018
M. F. Hasler, A050186, rows 0..100, Sep 27 2018
M. F. Hasler, A050186, rows 0..200, Sep 27 2018
N. J. A. Sloane, Transforms
Index entries for triangles and arrays related to Pascal's triangle

Same triangle as A053727 except this one includes column 0.

T(2n, n), T(2n+1, n) match A007727, A001700, respectively. Row sums match A027375.

T[n_, k_] := If[n == 0, 1, DivisorSum[GCD[k, n], MoebiusMu[#] Binomial[n/#, k/#]&]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 16 2022 *)

A050186(n,k)=sumdiv(gcd(n+!n,k),d,moebius(d)*binomial(n/d,k/d)) \\ M. F. Hasler, Sep 27 2018

MOEBIUS transform of A007318 Pascal's Triangle.

If rows n > 1 are divided by n, this yields the triangle A051168, which equals A245558 surrounded by 0's (except for initial terms). This differs from A011847 from row n = 9 on. - M. F. Hasler, Sep 29 2018

A051841 Number of binary Lyndon words with an even number of 1's.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 28, 48, 93, 165, 315, 576, 1091, 2032, 3855, 7252, 13797, 26163, 49929, 95232, 182361, 349350, 671088, 1290240, 2485504, 4792905, 9256395, 17894588, 34636833, 67106816, 130150493, 252641280, 490853403, 954429840, 1857283155, 3616800768, 7048151355, 13743869130, 26817356775

Offset: 1

Frank Ruskey, Dec 13 1999

Also number of trace 0 irreducible polynomials over GF(2).

Also number of trace 0 Lyndon words over GF(2).

			a(5) = 3 = |{ 00011, 00101, 01111 }|.

May, Robert M. "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019

T. D. Noe, Table of n, a(n) for n = 1..300
F. Ruskey, Number of q-ary Lyndon words with given trace mod q
F. Ruskey, Number of Lyndon words over GF(q) with given trace
Index entries for sequences related to Lyndon words

Same as A001037 - A000048. Same as A042980 + A042979.
Cf. A027750, A008683.
Cf. A000010.

a051841 n = (sum $ zipWith (\u v -> gcd 2 u * a008683 u * 2 ^ v)
             ds $ reverse ds) `div` (2 * n) where ds = a027750_row n
-- Reinhard Zumkeller, Mar 17 2013

a[n_] := Sum[GCD[d, 2]*MoebiusMu[d]*2^(n/d), {d, Divisors[n]}]/(2n);
Table[a[n], {n, 1, 32}]
(* Jean-François Alcover, May 14 2012, from formula *)

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==0, L(n, k), 0 ) ) / n;
vector(33,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

a(n) = 1/(2*n)*Sum_{d|n} gcd(d,2)*mu(d)*2^(n/d).
a(n) ~ 2^(n-1) / n. - Vaclav Kotesovec, May 31 2019
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = 1/(2*n)*Sum_{k=1..n} gcd(gcd(n,k),2)*mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} gcd(n/gcd(n,k),2)*mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)

A296658 Length of the standard Lyndon word factorization of the first n terms of A000002.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 18 2017

nonn

			The standard Lyndon word factorization of (12211212212211211) is (122)(112122122)(112)(1)(1), so a(17) = 5.
The standard factorizations of initial terms of A000002:
1
12
122
122,1
122,1,1
122,112
122,112,1
122,11212
122,112122
122,112122,1
122,11212212
122,112122122
122,112122122,1
122,112122122,1,1
122,112122122,112
122,112122122,112,1
122,112122122,112,1,1
122,112122122,112,112
122,112122122,1121122
122,112122122,1121122,1

Frédérique Bassino, Julien Clement, and Cyril Nicaud, The standard factorization of Lyndon words: an average point of view, Discrete Mathematics, 290-1 (2005), 1-25.

Row lengths of A329315.
The "co-" version is A329362.
Cf. A000002, A001037, A027375, A060223, A088568, A102659, A211100, A281013, A288605, A296372, A296657, A296659, A328596, A329312, A329316, A329317.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],LyndonQ[Take[q,#]]&]];
kolagrow[q_]:=If[Length[q]<2,Take[{1,2},Length[q]+1],Append[q,Switch[{q[[Length[Split[q]]]],Part[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]]];
Table[Length[qit[Nest[kolagrow,1,n]]],{n,150}]

A152061 Counts of unique periodic binary strings of length n.

Original entry on oeis.org

0, 0, 2, 2, 4, 2, 10, 2, 16, 8, 34, 2, 76, 2, 130, 38, 256, 2, 568, 2, 1036, 134, 2050, 2, 4336, 32, 8194, 512, 16396, 2, 33814, 2, 65536, 2054, 131074, 158, 266176, 2, 524290, 8198, 1048816, 2, 2113462, 2, 4194316, 33272, 8388610, 2, 16842496, 128, 33555424

Offset: 0

Jin S. Choi, Sep 24 2011

nonn

a(p) = 2 for p prime.

			a(3) = 2 = |{ 000, 111 }|, a(4) = 4 = |{ 0000, 1111, 0101, 1010 }|.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
Achilles A. Beros, Bjørn Kjos-Hanssen, and Daylan Kaui Yogi, Planar digraphs for automatic complexity, arXiv:1902.00812 [cs.FL], 2019.

Row sums of A050870.
A050871 is bisection (even part). - R. J. Mathar, Sep 24 2011
Cf. A008683, A178472.

with(numtheory):
a:= n-> `if`(n=0, 0, 2^n -add(mobius(n/d)*2^d, d=divisors(n))):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 26 2011

a[0] = 0; a[n_] := 2^n - Sum[MoebiusMu[n/d]*2^d, {d, Divisors[n]}];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jul 04 2019 *)

from sympy import mobius, divisors
def A152061(n): return -sum(mobius(n//d)<Chai Wah Wu, Sep 21 2024

a(n) = 2^n - A001037(n) * n for n>0, a(0) = 0.
a(n) = 2^n - A027375(n) for n>0, a(0) = 0.
a(n) = 2^n - Sum_{d|n} mu(n/d) 2^d for n>0, a(0) = 0.
a(n) = 2^n - A143324(n,2).
a(n) = 2 * A178472(n) for n > 0. - Alois P. Heinz, Jul 04 2019

A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

Original entry on oeis.org

2, 1, 1, 2, 2, 2, 3, 9, 9, 3, 6, 27, 54, 27, 6, 9, 99, 335, 335, 99, 9, 18, 326, 2182, 4050, 2182, 326, 18, 30, 1161, 14523, 52377, 52377, 14523, 1161, 30, 56, 4050, 99858, 698535, 1342170, 698535, 99858, 4050, 56, 99, 14532, 698870, 9586395, 35790267, 35790267, 9586395, 698870, 14532, 99

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional (Lyndon word) case is A001037.

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Table begins:
        1    2    3    4
    ------------------------
  1: |  2    1    2    3
  2: |  1    2    9   27
  3: |  2    9   54  335
  4: |  3   27  335 4050
Inequivalent representatives of the A(3,2) = 9 aperiodic toroidal necklaces:
  [0 0 0] [0 0 0] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 1 1] [0 1 1]
  [0 0 1] [0 1 1] [0 1 0] [0 1 1] [1 0 1] [1 1 0] [1 1 1] [1 0 1] [1 1 1]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.
Andrew Howroyd, GAP Program Code

First and last columns are A001037. Main diagonal is A323872.
Cf. A000031, A000740, A027375, A059966, A179043, A184271.
Cf. A323859, A323860, A323865, A323866, A323871.

# See link for code.
for n in [1..8] do for k in [1..8] do Print(A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],And[apermatQ[#],neckmatQ[#]]&]],{n,6},{k,n-1}]

Terms a(37) and beyond from Andrew Howroyd, Aug 21 2019

A323865 Number of aperiodic binary toroidal necklaces of size n.

Original entry on oeis.org

1, 2, 2, 4, 8, 12, 36, 36, 114, 166, 396, 372, 1992, 1260, 4644, 8728, 20310, 15420, 87174, 55188, 314064, 399432, 762228, 729444, 5589620, 4026522, 10323180, 19883920, 57516048, 37025580, 286322136, 138547332, 805277760, 1041203944, 2021145660, 3926827224

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

We define a toroidal necklace to be an equivalence class of matrices under all possible rotations of the sequence of rows and the sequence of columns. An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			Inequivalent representatives of the a(6) = 36 aperiodic necklaces:
  000001  000011  000101  000111  001011  001101  001111  010111  011111
.
  000  000  001  001  001  001  001  011  011
  001  011  010  011  101  110  111  101  111
.
  00  00  00  00  00  01  01  01  01
  00  01  01  01  11  01  01  10  11
  01  01  10  11  01  10  11  11  11
.
  0  0  0  0  0  0  0  0  0
  0  0  0  0  0  0  0  1  1
  0  0  0  0  1  1  1  0  1
  0  0  1  1  0  1  1  1  1
  0  1  0  1  1  0  1  1  1
  1  1  1  1  1  1  1  1  1

Andrew Howroyd, Table of n, a(n) for n = 0..200
S. N. Ethier, Counting toroidal binary arrays, J. Int. Seq. 16 (2013) #13.4.7.

Cf. A000031, A001037, A027375, A179043, A184271, A323351.

Cf. A323858, A323859, A323860, A323861, A323864, A323871.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
neckmatQ[m_]:=m==First[Union@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}]];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[If[n==0,1,Length[Union[First/@matcyc/@Select[zaz[n],And[apermatQ[#],neckmatQ[#]]&]]]],{n,0,10}]

a(n) = Sum_{d|n} A323861(d, n/d) for n > 0. - Andrew Howroyd, Aug 21 2019

Terms a(19) and beyond from Andrew Howroyd, Aug 21 2019

A329315 Irregular triangle read by rows where row n gives the sequence of lengths of components of the Lyndon factorization of the first n terms of A000002.

Original entry on oeis.org

1, 2, 3, 3, 1, 3, 1, 1, 3, 3, 3, 3, 1, 3, 5, 3, 6, 3, 6, 1, 3, 8, 3, 9, 3, 9, 1, 3, 9, 1, 1, 3, 9, 3, 3, 9, 3, 1, 3, 9, 3, 1, 1, 3, 9, 3, 3, 3, 9, 7, 3, 9, 7, 1, 3, 9, 9, 3, 9, 9, 1, 3, 9, 9, 1, 1, 3, 9, 9, 3, 3, 9, 9, 3, 1, 3, 9, 14, 3, 9, 15, 3, 9, 15, 1, 3

Offset: 1

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 4) never appear in the sequence.

			Triangle begins:
   1: (1)
   2: (2)
   3: (3)
   4: (3,1)
   5: (3,1,1)
   6: (3,3)
   7: (3,3,1)
   8: (3,5)
   9: (3,6)
  10: (3,6,1)
  11: (3,8)
  12: (3,9)
  13: (3,9,1)
  14: (3,9,1,1)
  15: (3,9,3)
  16: (3,9,3,1)
  17: (3,9,3,1,1)
  18: (3,9,3,3)
  19: (3,9,7)
  20: (3,9,7,1)
For example, the first 10 terms of A000002 are (1221121221), with Lyndon factorization (122)(112122)(1), so row 10 is (3,6,1).

Row lengths are A296658.
The reversed version is A329316.
Cf. A000002, A000031, A001037, A027375, A059966, A060223, A088568, A102659, A211100, A288605, A296372, A329314, A329317, A329325.

lynQ[q_]:=Array[Union[{q,RotateRight[q,#1]}]=={q,RotateRight[q,#1]}&,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]]&]]]];
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[kol[n]],{n,100}]

A038199 Row sums of triangle T(m,n) = number of solutions to 1 <= a(1) < a(2) < ... < a(m) <= n, where gcd(a(1), a(2), ..., a(m), n) = 1, in A020921.

Original entry on oeis.org

1, 2, 6, 12, 30, 54, 126, 240, 504, 990, 2046, 4020, 8190, 16254, 32730, 65280, 131070, 261576, 524286, 1047540, 2097018, 4192254, 8388606, 16772880, 33554400, 67100670, 134217216, 268419060, 536870910, 1073708010, 2147483646

Offset: 1

Temba Shonhiwa (Temba(AT)maths.uz.ac.zw)

The function T(m,n) described above has an inverse: see A038200.

Also, Moebius transform of 2^n - 1 = A000225. Also, number of rationals in [0, 1) whose binary expansions consist just of repeating bits of (least) period exactly n (i.e., there's no preperiodic part), where 0 = 0.000... is considered to have period 1. - Brad Chalfan (brad(AT)chalfan.net), May 29 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Henk Bruin, Carlo Carminati, and Charlene Kalle, Matching for generalised beta-transformations, arXiv preprint arXiv:1610.01872 [math.DS], 2016.
Henk Bruin, Carlo Carminati, and Charlene Kalle, Matching for generalised beta-transformations, Indagationes Mathematicae 28 (2017), 55-73.
Jason D. Chadwick, Mariesa H. Teo, Joshua Viszlai, Willers Yang, and Frederic T. Chong, Erasure Minesweeper: exploring hybrid-erasure surface code architectures for efficient quantum error correction, arXiv:2505.00066 [quant-ph], 2025. See p. 14.
Melvyn B. Nathanson, Primitive sets and Euler phi function for subsets of {1,2,...,n}, arXiv:math/0608150 [math.NT], 2006-2007.
Prapanpong Pongsriiam, Relatively Prime Sets, Divisor Sums, and Partial Sums, arXiv:1306.4891 [math.NT], 2013 and J. Int. Seq. 16 (2013) #13.9.1.
Prapanpong Pongsriiam, A remark on relatively prime sets, Integers 13 (2013), A49.
Temba Shonhiwa, A Generalization of the Euler and Jordan Totient Functions, Fib. Quart., 37 (1999), 67-76.
Wikipedia, Lambert series.

A027375, A038199 and A056267 are all essentially the same sequence with different initial terms.
Cf. A000225, A008683, A020921, A023995, A027750, A038200, A130887.
Cf. A059966 (a(n)/n).

a038199 n = sum [a008683 (n `div` d) * (a000225 d)| d <- a027750_row n]
-- Reinhard Zumkeller, Feb 17 2013

Table[Plus@@((2^Divisors[n]-1)MoebiusMu[n/Divisors[n]]),{n,1,31}] (* Brad Chalfan (brad(AT)chalfan.net), May 29 2006 *)

a(n) = sumdiv(n, d, moebius(n/d)*(2^d-1)); \\ Michel Marcus, Jun 28 2017

from sympy import mobius, divisors
def a(n): return sum(mobius(n//d) * (2**d - 1) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jun 28 2017

a(n) = Sum_{d | n} mu(n/d)*(2^d-1). - Paul Barry, Mar 20 2005
Lambert g.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/((1 - x)*(1 - 2*x)). - Ilya Gutkovskiy, Apr 25 2017
O.g.f.: Sum_{d >= 1} mu(d)*x^d/((1 - x^d)*(1 - 2*x^d)). - Petros Hadjicostas, Jun 18 2019

Better description from Michael Somos
More terms from Naohiro Nomoto, Sep 10 2001
More terms from Brad Chalfan (brad(AT)chalfan.net), May 29 2006

A323860 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary arrays.

Original entry on oeis.org

2, 2, 2, 6, 8, 6, 12, 54, 54, 12, 30, 216, 486, 216, 30, 54, 990, 4020, 4020, 990, 54, 126, 3912, 32730, 64800, 32730, 3912, 126, 240, 16254, 261414, 1047540, 1047540, 261414, 16254, 240, 504, 64800, 2097018, 16764840, 33554250, 16764840, 2097018, 64800, 504

Offset: 1

Gus Wiseman, Feb 04 2019

The 1-dimensional case is A027375.

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

			Table begins:
       1     2     3     4
    ------------------------
  1: |  2     2     6    12
  2: |  2     8    54   216
  3: |  6    54   486  4020
  4: | 12   216  4020 64800
The A(2,2) = 8 arrays:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are independently aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

First and last columns are A027375. Main diagonal is A323863.
Cf. A000740, A001037, A179043, A265627, A323351.
Cf. A323861, A323862, A323864, A323865, A323867, A323869.

# See A323861 for code.
for n in [1..8] do for k in [1..8] do Print(n*k*A323861(n,k), ", "); od; Print("\n"); od; # Andrew Howroyd, Aug 21 2019

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[Partition[#,n-k]&/@Tuples[{0,1},(n-k)*k],apermatQ]],{n,8},{k,n-1}]

T(n,k) = n*k*A323861(n,k). - Andrew Howroyd, Aug 21 2019

Terms a(29) and beyond from Andrew Howroyd, Aug 21 2019

cp's OEIS Frontend

A138904 Number of rotational symmetries in the binary expansion of a number.

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A050186 Triangular array T read by rows: T(h,k) = number of binary words of k 1's and h-k 0's which are not a juxtaposition of 2 or more identical subwords.

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A051841 Number of binary Lyndon words with an even number of 1's.

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A296658 Length of the standard Lyndon word factorization of the first n terms of A000002.

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A152061 Counts of unique periodic binary strings of length n.

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A323861 Table read by antidiagonals where A(n,k) is the number of n X k aperiodic binary toroidal necklaces.

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A323865 Number of aperiodic binary toroidal necklaces of size n.

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