A058766 -id:A058766 - cp's OEIS frontend

A058821 Dimensions of homogeneous subspaces of shuffle algebra over 6-letter alphabet (see A058766 for 2-letter case).

1, 6, 21, 146, 981, 6222, 38921, 239946, 1469826, 8957976, 54420339, 329815506, 1995387801, 12056025246, 72766743801, 438839319470, 2644790643216, 15930973595046, 95917737415956, 577288174746786, 3473350521083199, 20892333943230346, 125638899138654861

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A000400, A008683, A032164, A058766.

a[n_] := 6^n - DivisorSum[n, MoebiusMu[n/#] * 6^# &] / n; a[0] = 1; a[1] = 6; Array[a, 23, 0] (* Amiram Eldar, Aug 13 2023 *)

For n >= 2, a(n) = 6^n - (1/n) * Sum_{d|n} A008683(n/d) * 6^d. - Sean A. Irvine, Aug 28 2022

a(n) = 6^n - A032164(n) for n >= 2. - Amiram Eldar, Aug 13 2023

More terms from Sean A. Irvine, Aug 28 2022

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352, 632, 1182, 2192, 4116, 7712, 14602, 27596, 52488, 99880, 190746, 364724, 699252, 1342184, 2581428, 4971068, 9587580, 18512792, 35792568, 69273668, 134219796, 260301176, 505294128, 981706832

Offset: 0

N. J. A. Sloane

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011
Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019
(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021
From Jianing Song, Nov 13 2021: (Start)
a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.
a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).
a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

			For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.
The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111...}.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.
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
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, pp. 379-383.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
J.-M. Champarnaud, G. Hansel, and D. Perrin, Unavoidable sets of constant length, Internat. J. Alg. Comput. 14 (2004), 241-251.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
S. N. Ethier and J. Lee, Parrondo games with spatial dependence, arXiv preprint arXiv:1202.2609 [math.PR], 2012.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see pages 18, 64.
H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.
Harold Fredricksen, An algorithm for generating necklaces of beads in two colors, Discrete Mathematics, Volume 61, Issues 2-3, September 1986, Pages 181-188.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 2
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 130
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
Abraham Lempel, On extremal factors of the de Bruijn graph, J. Combinatorial Theory Ser. B 11 1971 17--27. MR0276126 (43 #1874).
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Johannes Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combinatorial Theory Ser. B 13 (1972), 40-45. MR0323629 (48 #1985).
Matthew Parker, The first 25K terms (7-Zip compressed file) [a large file]
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Ville Salo, Universal gates with wires in a row, arXiv:1809.08050 [math.GR], 2018.
J. A. Siehler, The Finite Lamplighter Groups: A Guided Tour, College Mathematics Journal, Vol. 43, No. 3 (May 2012), pp. 203-211.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
David Thomson, Musical Polygons, Mathematics Today, Vol. 57, No. 2 (April 2021), pp. 50-51.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012.
Eric Weisstein's World of Mathematics, Necklace
Wolfram Research, Number of necklaces
Index entries for "core" sequences
Index entries for sequences related to necklaces

Column 2 of A075195.
Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.
Rows sums of triangle in A047996.
Dividing by 2 gives A053634.
A008965(n) = a(n) - 1 allowing different offsets.
Cf. A008965, A053635, A052823, A100447 (bisection).
Cf. A000010.

a000031 0 = 1
a000031 n = (`div` n) $ sum $
   zipWith (*) (map a000010 divs) (map a000079 $ reverse divs)
   where divs = a027750_row n
-- Reinhard Zumkeller, Mar 21 2013

with(numtheory); A000031 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];

a[n_] := Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n/d), 0], {d, 1, n}]/n
a[n_] := Fold[#1 + 2^(n/#2) EulerPhi[#2] &, 0, Divisors[n]]/n (* Ben Branman, Jan 08 2011 *)
Table[Expand[CycleIndex[CyclicGroup[n], t] /. Table[t[i]-> 2, {i, 1, n}]], {n,0, 30}] (* Geoffrey Critzer, Mar 06 2011*)
a[0] = 1; a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2016 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-2*x^i]/i,{i,1,mx}],{x,0,mx}],x] (*Herbert Kociemba, Oct 29 2016 *)

{A000031(n)=if(n==0,1,sumdiv(n,d,eulerphi(d)*2^(n/d))/n)} \\ Randall L Rathbun, Jan 11 2002

from sympy import totient, divisors
def A000031(n): return sum(totient(d)*(1<Chai Wah Wu, Nov 16 2022

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d) = A053635(n)/n, where phi is A000010.
Warning: easily confused with A001037, which has a similar formula.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n. - Herbert Kociemba, Oct 29 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021
Dirichlet g.f.: f(s+1) * (zeta(s)/zeta(s+1)), where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).

A091242 Reducible polynomials over GF(2), coded in binary.

Original entry on oeis.org

4, 5, 6, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 26, 27, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 88

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

"Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1). - M. F. Hasler, Aug 18 2014
The reducible polynomials in GF(2)[X] are the analog to the composite numbers A002808 in the integers.
It follows that the sequence is closed under application of A048720(.,.), which effects multiplication of the coded polynomials. It is also closed under application of blue code, A193231. The majority of the terms are coded multiples of X^1 (represented by 2) and/or X^1+1 (represented by 3): see A005843 and A001969 respectively. A246157 lists the other terms. - Peter Munn, Apr 20 2021

			For example, 5 = 101 in binary encodes the polynomial x^2+1 which is factored as (x+1)^2 in the polynomial ring GF(2)[X].

Robert Israel, Table of n, a(n) for n = 1..10000
Antti Karttunen, Scheme-program for computing this sequence.
Index entries for sequences operating on GF(2)[X]-polynomials

Inverse: A091246. Almost complement of A014580. Union of A091209 & A091212. First differences: A091243. Characteristic function: A091247. In binary format: A091254.
Number of degree-n reducible polynomials: A058766.
Subsequences: A001969\{0,3}, A005843\{0,2}, A246156, A246157, A246158, A316970.
Cf. A002808, A048720, A193231.

filter:= proc(n) local L;
  L:= convert(n,base,2);
  not Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2
end proc:
select(filter, [$2..200]); # Robert Israel, Aug 30 2018

okQ[n_] := Module[{x, id = IntegerDigits[n, 2] // Reverse}, !IrreduciblePolynomialQ[id.x^Range[0, Length[id]-1], Modulus -> 2]];
Select[Range[2, 200], okQ] (* Jean-François Alcover, Jan 04 2022 *)

Edited by M. F. Hasler, Aug 18 2014

A058818 a(0) = 1, a(1) = 3; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(3), i.e., a(n) = 3^n - A027376(n).

Original entry on oeis.org

1, 3, 6, 19, 63, 195, 613, 1875, 5751, 17499, 53169, 161043, 487221, 1471683, 4441485, 13392331, 40356711, 121543683, 365898261, 1101089811, 3312448137, 9962241251, 29954655861, 90049997139, 270661661541, 813397065075, 2444101819329, 7343167949235

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 3-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A027376, A058766.

a[n_] := 3^n - DivisorSum[n, MoebiusMu[n/#] * 3^# &] / n; a[0] = 1; a[1] = 3; Array[a, 28, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 3^n, 3^n - sumdiv(n, d, moebius(d)*3^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

a(16)-a(27) from Alois P. Heinz, Nov 25 2016

A058819 a(0) = 1, a(1) = 4; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(4), i.e., a(n) = 4^n - A027377(n).

Original entry on oeis.org

1, 4, 10, 44, 196, 820, 3426, 14044, 57376, 233024, 943822, 3813004, 15379476, 61946644, 249262666, 1002159108, 4026535936, 16169288644, 64901742816, 260410648684, 1044536098828, 4188615725644, 16792541414866, 67309233561244, 269746853382816, 1080863910568960, 4330384259668126

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 4-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A000302, A027377, A058766.

a[n_] := 4^n - DivisorSum[n, MoebiusMu[n/#] * 4^# &] / n; a[0] = 1; a[1] = 4; Array[a, 27, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 4^n, 4^n - sumdiv(n, d, moebius(d)*4^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

A058820 a(0) = 1, a(1) = 5; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(5), i.e., a(n) = 5^n - A001692(n).

Original entry on oeis.org

1, 5, 15, 85, 475, 2501, 13045, 66965, 341875, 1736125, 8789377, 44389205, 223796925, 1126802885, 5667555805, 28483073133, 143051171875, 718060661765, 3602769749125, 18069618626645, 90599060546905, 454130626863845, 2275813711825285, 11402627696161685, 57121117919938125

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 5-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A000351, A001692, A058766.

a[n_] := 5^n - DivisorSum[n, MoebiusMu[n/#] * 5^# &] / n; a[0] = 1; a[1] = 5; Array[a, 25, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 5^n, 5^n - sumdiv(n, d, moebius(d)*5^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

A058822 a(0) = 1, a(1) = 7; for n>=2 a(n) is the number of degree-n monic reducible polynomials over GF(7), i.e., a(n) = 7^n - A001693(n).

Original entry on oeis.org

1, 7, 28, 231, 1813, 13447, 98105, 705895, 5044501, 35869911, 254229409, 1797569767, 12687856601, 89436009607, 629778626473, 4431057410423, 31155872769301, 218946366105607, 1537946178052697, 10798953333511399, 75802652996855281, 531948441984119239, 3732101910100912537

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 7-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A000420, A001693, A058766.

a[n_] := 7^n - DivisorSum[n, MoebiusMu[n/#] * 7^# &] / n; a[0] = 1; a[1] = 7; Array[a, 23, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 7^n, 7^n - sumdiv(n, d, moebius(d)*7^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

A058823 a(0) = 1, a(1) = 8; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(8), i.e., a(n) = 8^n - A027380(n).

Original entry on oeis.org

1, 8, 36, 344, 3088, 26216, 218548, 1797560, 14680576, 119304704, 966370924, 7809031448, 62992875856, 507466905128, 4083900481540, 32838747285128, 263882791714816, 2119341001115528, 17013598599759616, 136530178177126616, 1095275429430191920, 8784163844623695896

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 8-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A001018, A027380, A058766.

a[n_] := 8^n - DivisorSum[n, MoebiusMu[n/#] * 8^# &] / n; a[0] = 1; a[1] = 8; Array[a, 22, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 8^n, 8^n - sumdiv(n, d, moebius(d)*8^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

A058824 a(0) = 1, a(1) = 9; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(9), i.e., a(n) = 9^n - A027381(n).

Original entry on oeis.org

1, 9, 45, 489, 4941, 47241, 443001, 4099689, 37666701, 344373849, 3138111873, 28528236009, 258893786601, 2346337687689, 21242736192681, 192165056625657, 1737206429739021, 15696171011450889, 141756044468718681, 1279754258848097769, 11549782186278421905, 104208561077631046089

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Jan 04 2001

nonn

Dimensions of homogeneous subspaces of shuffle algebra over 9-letter alphabet (see A058766 for 2-letter case).

M. Lothaire, Combinatorics on words, Cambridge mathematical library, 1983, p. 126 (definition of shuffle algebra).

Cf. A001019, A027381, A058766.

a[n_] := 9^n - DivisorSum[n, MoebiusMu[n/#] * 9^# &] / n; a[0] = 1; a[1] = 9; Array[a, 22, 0] (* Amiram Eldar, Aug 13 2023 *)

a(n) = if (n<=1, 9^n, 9^n - sumdiv(n, d, moebius(d)*9^(n/d))/n); \\ Michel Marcus, Oct 30 2017

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Feb 19 2002

More terms from Michel Marcus, Oct 30 2017

cp's OEIS Frontend

A058821 Dimensions of homogeneous subspaces of shuffle algebra over 6-letter alphabet (see A058766 for 2-letter case).

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A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

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A091242 Reducible polynomials over GF(2), coded in binary.

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A058818 a(0) = 1, a(1) = 3; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(3), i.e., a(n) = 3^n - A027376(n).

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Mathematica

PARI

Extensions

A058819 a(0) = 1, a(1) = 4; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(4), i.e., a(n) = 4^n - A027377(n).

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Crossrefs

Programs

Mathematica

PARI

Extensions

A058820 a(0) = 1, a(1) = 5; for n >= 2 a(n) is the number of degree-n monic reducible polynomials over GF(5), i.e., a(n) = 5^n - A001692(n).

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Comments

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Crossrefs

Programs

Mathematica

PARI

Extensions

A058822 a(0) = 1, a(1) = 7; for n>=2 a(n) is the number of degree-n monic reducible polynomials over GF(7), i.e., a(n) = 7^n - A001693(n).

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