A054719 -id:A054719 - cp's OEIS frontend

A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

1, 2, 0, 3, 2, 0, 4, 6, 6, 0, 5, 12, 24, 12, 0, 6, 20, 60, 72, 30, 0, 7, 30, 120, 240, 240, 54, 0, 8, 42, 210, 600, 1020, 696, 126, 0, 9, 56, 336, 1260, 3120, 4020, 2184, 240, 0, 10, 72, 504, 2352, 7770, 15480, 16380, 6480, 504, 0, 11, 90, 720, 4032, 16800, 46410, 78120, 65280, 19656, 990, 0

Offset: 1

Alois P. Heinz, Aug 07 2008

Column k is Dirichlet convolution of mu(n) with k^n.

The coefficients of the polynomial of row n are given by the n-th row of triangle A054525; for example row 4 has polynomial -k^2+k^4.

			T(2,3)=6, because there are 6 primitive words of length 2 over 3-letter alphabet {a,b,c}: ab, ac, ba, bc, ca, cb; note that the non-primitive words aa, bb and cc don't belong to the list; secondly note that the words in the list need not be Lyndon words, for example ba can be derived from ab by a cyclic rotation of the positions.
Table begins:
  1,  2,   3,    4,    5, ...
  0,  2,   6,   12,   20, ...
  0,  6,  24,   60,  120, ...
  0, 12,  72,  240,  600, ...
  0, 30, 240, 1020, 3120, ...

Alois P. Heinz, Antidiagonals n = 1..141, flattened
C. J. Smyth, A coloring proof of a generalisation of Fermat's little theorem, Amer. Math. Monthly 93, No. 6 (1986), pp. 469-471.
Lucas Vieira Barbosa, Nonclassical Temporal Correlations Under Finite-Memory Constraints, Doctoral Thesis, Univ. Wien (Austria 2024). See p. 35.
Index entries for sequences related to Lyndon words

Columns k=1-10 give: A000007(n-1), A027375, A054718, A054719, A054720, A054721, A218124, A218125, A218126, A218127.
Rows n=1-10 give: A000027, A002378(k-1), A007531(k+1), A047928(k+1), A061167, A218130, A133499, A218131, A218132, A218133.
Main diagonal gives A252764.
Cf. A074650, A143325, A008683, A054525.

with(numtheory): f0:= proc(n) option remember; unapply(k^n-add(f0(d)(k), d=divisors(n)minus{n}), k) end; T:= (n,k)-> f0(n)(k); seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

f0[n_] := f0[n] = Function [k, k^n - Sum[f0[d][k], {d, Complement[Divisors[n], {n}]}]]; t[n_, k_] := f0[n][k]; Table[Table[t[n, 1 + d - n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = Sum_{d|n} k^d * mu(n/d).

T(n,k) = k^n - Sum_{d

T(n,k) = A143325(n,k) * k.

T(n,k) = A074650(n,k) * n.

So Sum_{d|n} k^d * mu(n/d) == 0 (mod n), this is a generalization of Fermat's little theorem k^p - k == 0 (mod p) for primes p to an arbitrary modulus n (see the Smyth link). - Franz Vrabec, Feb 09 2021

A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

Original entry on oeis.org

1, 4, 6, 20, 60, 204, 670, 2340, 8160, 29120, 104754, 381300, 1397740, 5162220, 19172790, 71582716, 268431360, 1010580540, 3817733920, 14467258260, 54975528948, 209430785460, 799644629550, 3059510616420

Offset: 0

N. J. A. Sloane

Apart from initial terms, exponents in expansion of A065419 as a product zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 4 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.

Seiichi Manyama, Table of n, a(n) for n = 0..1666 (terms 0..200 from T. D. Noe)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
A. Pakapongpun and T. Ward, Functorial Orbit counting, JIS 12 (2009) 09.2.4, example 3.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
G. Viennot, Algèbres de Lie Libres et Monoïdes Libres, Lecture Notes in Mathematics 691, Springer Verlag 1978.
Index entries for sequences related to Lyndon words

Cf. A001037, A027376, A054719, A054660, A054661.

Column k=4 of A074650.

A027377 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s/n); fi; end;

a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}] / n; a[0] = 1; Table[a[n], {n, 0, 23}](* Jean-François Alcover, Nov 29 2011 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,4],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)<<(2*n/d))/n,1) \\ Charles R Greathouse IV, Nov 29 2011

a(n) = Sum_{d|n} mu(d)*4^(n/d)/n.
G.f.: k=4, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016
a(n) = A054661(n) + 3 * A054660(n). - Andrey Zabolotskiy, Dec 17 2020
a(n) = 2 * (A054664(n) + A054660(n)). - Andrey Zabolotskiy, Dec 19 2020
a(n) = A054719(n)/n, n>0. - R. J. Mathar, Dec 16 2024

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

A295505 a(n) = Sum_{d|n} mu(n/d)*4^(d-1).

Original entry on oeis.org

1, 3, 15, 60, 255, 1005, 4095, 16320, 65520, 261885, 1048575, 4193220, 16777215, 67104765, 268435185, 1073725440, 4294967295, 17179802640, 68719476735, 274877644740, 1099511623665, 4398045462525, 17592186044415, 70368739967040, 281474976710400

Offset: 1

Seiichi Manyama, Nov 23 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1661

Sum_{d|n} mu(n/d)*k^(d-1): A000740 (k=2), A034741 (k=3), this sequence (k=4), A295506 (k=5).
Column k=4 of A143325.
First differences of A320088.
Cf. A054719.

Table[Sum[MoebiusMu[n/d]4^(d-1),{d,Divisors[n]}],{n,30}] (* Harvey P. Dale, Nov 08 2020 *)
nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] * x^k / (1 - 4*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

{a(n) = sumdiv(n, d, moebius(n/d)*4^(d-1))}

a(n) = A054719(n)/4 for n > 0.

G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 4*x^k). - Ilya Gutkovskiy, Oct 25 2018

A056269 Number of primitive (aperiodic) words of length n which contain exactly four different symbols.

Original entry on oeis.org

0, 0, 0, 24, 240, 1560, 8400, 40800, 186480, 818280, 3498000, 14674440, 60780720, 249393480, 1016542560, 4123132800, 16664094960, 67171179600, 270232006800, 1085569963080, 4356217672800, 17466683473800

Offset: 1

Marks R. Nester

nonn

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]

Cf. A000302, A000919, A054719.

sum mu(d)*A000919(n/d) where d|n.

Seems to be n * A056289.

A056275 Number of primitive (aperiodic) word structures of length n using a 4-ary alphabet.

Original entry on oeis.org

1, 1, 4, 13, 50, 181, 714, 2780, 11046, 43895, 175274, 699875, 2798250, 11188191, 44747380, 178970560, 715860650, 2863365834, 11453377194, 45813202675, 183252461532, 733008625151, 2932033104554, 11728127521060, 46912504507000, 187649998452735, 750599971438464

Offset: 1

Marks R. Nester

nonn

Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.

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]

Cf. A008683, A054719.

a(n) = Sum_{d|n} mu(d)*A007581(n/d-1).

A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 2, 3, 1, 0, 0, 6, 6, 4, 1, 0, 0, 12, 24, 12, 5, 1, 0, 0, 30, 72, 60, 20, 6, 1, 0, 0, 54, 240, 240, 120, 30, 7, 1, 0, 0, 126, 696, 1020, 600, 210, 42, 8, 1, 0, 0, 240, 2184, 4020, 3120, 1260, 336, 56, 9, 1

Offset: 0

Peter Luschny, Jul 04 2023

Row n gives the number of n-ary sequences with primitive period k.

See A074650 and A143324 for combinatorial interpretations.

			Array A(n, k) starts:
[0] 1, 0,  0,   0,    0,     0,      0,       0,        0, ... A000007
[1] 1, 1,  0,   0,    0,     0,      0,       0,        0, ... A019590
[2] 1, 2,  2,   6,   12,    30,     54,     126,      240, ... A027375
[3] 1, 3,  6,  24,   72,   240,    696,    2184,     6480, ... A054718
[4] 1, 4, 12,  60,  240,  1020,   4020,   16380,    65280, ... A054719
[5] 1, 5, 20, 120,  600,  3120,  15480,   78120,   390000, ... A054720
[6] 1, 6, 30, 210, 1260,  7770,  46410,  279930,  1678320, ... A054721
[7] 1, 7, 42, 336, 2352, 16800, 117264,  823536,  5762400, ... A218124
[8] 1, 8, 56, 504, 4032, 32760, 261576, 2097144, 16773120, ... A218125
A000012|A002378| A047928   |   A218130     |      A218131
    A001477,A007531,    A061167,        A133499,   (diagonal A252764)
.
Triangle T(n, k) starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,  1;
[3] 0, 0,  2,   1;
[4] 0, 0,  2,   3,   1;
[5] 0, 0,  6,   6,   4,   1;
[6] 0, 0, 12,  24,  12,   5,  1;
[7] 0, 0, 30,  72,  60,  20,  6, 1;
[8] 0, 0, 54, 240, 240, 120, 30, 7, 1;

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Variant: A143324.
Rows: A000007 (n=0), A019590 (n=1), A027375 (n=2), A054718 (n=3), A054719 (n=4), A054720, A054721, A218124, A218125.
Columns: A000012 (k=0), A001477 (k=1), A002378 (k=2), A007531(k=3), A047928, A061167, A218130, A133499, A218131.
Cf. A252764 (main diagonal), A074650, A363914.

A363916 := (n, k) -> local d; add(A363914(k, d) * n^d, d = 0 ..k):
for n from 0 to 9 do seq(A363916(n, k), k = 0..8) od;

def A363916(n, k): return sum(A363914(k, d) * n^d for d in range(k + 1))
for n in range(9): print([A363916(n, k) for k in srange(9)])
def T(n, k): return A363916(k, n - k)

If k > 0 then k divides A(n, k), see the transposed array of A074650.

If k > 0 then n divides A(n, k), see the transposed array of A143325.

Showing 1-7 of 7 results.

cp's OEIS Frontend

A143324 Table T(n,k) by antidiagonals. T(n,k) is the number of length n primitive (=aperiodic or period n) k-ary words (n,k >= 1).

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A027377 Number of irreducible polynomials of degree n over GF(4); dimensions of free Lie algebras.

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A054718 Number of ternary sequences with primitive period n.

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A295505 a(n) = Sum_{d|n} mu(n/d)*4^(d-1).

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A056269 Number of primitive (aperiodic) words of length n which contain exactly four different symbols.

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A056275 Number of primitive (aperiodic) word structures of length n using a 4-ary alphabet.

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A363916 Array read by descending antidiagonals. A(n, k) = Sum_{d=0..k} A363914(k, d) * n^d.

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