A054720 -id:A054720 - cp's OEIS frontend

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

1, 5, 10, 40, 150, 624, 2580, 11160, 48750, 217000, 976248, 4438920, 20343700, 93900240, 435959820, 2034504992, 9536718750, 44878791360, 211927516500, 1003867701480, 4768371093720, 22706531339280

Offset: 0

N. J. A. Sloane

Exponents in expansion of Hardy-Littlewood constant C_5 = 0.409874885.. = A269843 as a product_{n>=2} zeta(n)^(-a(n)).

Number of aperiodic necklaces with n beads of 5 colors. - Herbert Kociemba, Nov 25 2016

E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 84.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003
M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 79.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..1435 (terms 0..200 from T. D. Noe)
Alin Bostan, Alexander Marynych, and Kilian Raschel, On the least common multiple of several random~integers, arXiv:1901.03002 [math.PR], 2019.
Jeremie Detrey, P. J. Spaenlehauer, and P. Zimmermann, Computing the rho constant, Preprint 2016.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Gareth A. Jones and Alexander K. Zvonkin, Groups of prime degree and the Bateman-Horn conjecture, 2021.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Dionisel Y. Regalado and Rodel Azura, An Analytic Approximation to the Density of Twin Primes, Recoletos Multidisciplinary Research Journal (2019) Vol. 6, No. 2.
G. J. Simmons, The number of irreducible polynomials of degree n over GF(p), Amer. Math. Monthly, 77 (1970), 743-745.
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, A054720, A002105.
5th column of A074650. - Alois P. Heinz, Aug 08 2008
Cf. A008683, A000351, A027750.

a001692 n = flip div n $ sum $
            zipWith (*) (map a008683 divs) (map a000351 $ reverse divs)
            where divs = a027750_row n
-- Reinhard Zumkeller, Oct 07 2015

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*5^(n/d)/n, {d, Divisors[n]}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Mar 11 2014 *)
mx=40;f[x_,k_]:=1-Sum[MoebiusMu[i] Log[1-k*x^i]/i,{i,1,mx}];CoefficientList[Series[f[x,5],{x,0,mx}],x] (* Herbert Kociemba, Nov 25 2016 *)

a(n)=if(n,sumdiv(n,d,moebius(d)*5^(n/d))/n,1) \\ Charles R Greathouse IV, Jun 15 2011

a(n) = Sum_{d|n} mu(d)*5^(n/d)/n, for n>0.

G.f.: k=5, 1 - Sum_{i>=1} mu(i)*log(1 - k*x^i)/i. - Herbert Kociemba, Nov 25 2016

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).

Original entry on oeis.org

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

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

A295506 a(n) = Sum_{d|n} mu(n/d)*5^(d-1).

Original entry on oeis.org

1, 4, 24, 120, 624, 3096, 15624, 78000, 390600, 1952496, 9765624, 48824880, 244140624, 1220687496, 6103514976, 30517500000, 152587890624, 762939059400, 3814697265624, 19073484374880, 95367431624976, 476837148437496, 2384185791015624, 11920928906172000

Offset: 1

Seiichi Manyama, Nov 23 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..1431

Sum_{d|n} mu(n/d)*k^(d-1): A000740 (k=2), A034741 (k=3), A295505 (k=4), this sequence (k=5).
Column k=5 of A143325.
First differences of A320089.
Cf. A054720.

nmax = 20; Rest[CoefficientList[Series[Sum[MoebiusMu[k] * x^k / (1 - 5*x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Dec 11 2020 *)

{a(n) = sumdiv(n, d, moebius(n/d)*5^(d-1))}

a(n) = A054720(n)/5 for n > 0.

G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 5*x^k). - Ilya Gutkovskiy, Oct 25 2018

A056270 Number of primitive (aperiodic) words of length n which contain exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 5102880, 29607600, 165526200, 901020120, 4808987400, 25292030280, 131542740000, 678330198120, 3474970629480, 17710714165200, 89904725757000

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. A000351, A001118, A054720.

Sum mu(d)*A001118(n/d) where d|n.

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.

A056276 Number of primitive (aperiodic) word structures of length n using a 5-ary alphabet.

Original entry on oeis.org

1, 1, 4, 13, 51, 196, 854, 3830, 17997, 86419, 422004, 2079260, 10306751, 51263086, 255514299, 1275160060, 6368612301, 31821454413, 159042661904, 795019250650, 3974515029793, 19870830290476, 99348921288654, 496728909635860, 2483597478617750, 12417846151236799

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, A054720.

a(n) = Sum_{d|n} mu(d)*A056272(n/d).

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cp's OEIS Frontend

A001692 Number of irreducible polynomials of degree n over GF(5); dimensions of free Lie algebras.

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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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A054718 Number of ternary sequences with primitive period n.

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A295506 a(n) = Sum_{d|n} mu(n/d)*5^(d-1).

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A056270 Number of primitive (aperiodic) words of length n which contain exactly five different symbols.

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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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A056276 Number of primitive (aperiodic) word structures of length n using a 5-ary alphabet.

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