A011260 -id:A011260 - cp's OEIS frontend

A346019 Number of n X n invertible matrices over GF(2) that have order 2^n-1.

1, 2, 48, 2688, 1935360, 1919877120, 23222833643520, 335564785519165440, 65717007596073359769600, 21492090164219831579049984000, 66041307304745851496871108594892800, 226523509196861965428709270554756199219200, 16622838761287803491875715175557341313583022080000

Offset: 1

Geoffrey Critzer, Jul 01 2021

nonn

Equivalently, a(n) is the number of n X n matrices over GF(2) whose characteristic polynomial is primitive.

2^n - 1 is the greatest order that a matrix in the general linear group GL_n(F_2) can have.

M. R. Darafsheh, Order of elements in the groups related to the general linear group, Finite fields and their applications, 11 (2005), 738-747.

Cf. A000010, A011260, A002884, A000225, A345463.

a:= n-> mul(2^n-2^i, i=0..n-1)*numtheory[phi](2^n-1)/((2^n-1)*n):
seq(a(n), n=1..14);  # Alois P. Heinz, Jul 01 2021

nn = 13; Table[EulerPhi[2^n - 1]/n, {n, 1, nn}]* Table[Product[2^n - 2^i, {i, 0, n - 1}], {n, 1, nn}]/Table[2^n - 1, {n, 1, nn}]

a(n) = A011260(n) * A002884(n)/A000225(n).

A119512 Determinant of n X n matrix of first n^2 terms of A000020 number of primitive polynomials of degree n over GF(2).

Original entry on oeis.org

2, 2, 244, -80544, 2895473496576

Offset: 1

Jonathan Vos Post, May 27 2006

The initial 2 should probably be a 1, see: A011260. This would change all terms to: a(2) = 0 because of the singular determinant[1,1,2,2] = 0; a(3) = 52; a(4) = -34848; a(5) = -2211008492544.

			a(2) = 2 =
|2 1|
|2 2|.

Cf. A000020, A011260, A119493.

A173270 Partial sums of A001037, the number of degree-n irreducible polynomials over GF(2).

Original entry on oeis.org

1, 3, 4, 6, 9, 15, 24, 42, 72, 128, 227, 413, 748, 1378, 2539, 4721, 8801, 16511, 31043, 58637, 111014, 210872, 401429, 766151, 1465021, 2807197, 5387992, 10359000, 19945395, 38458185, 74248452, 143522118, 277737798, 538038784, 1043325199

Offset: 0

Jonathan Vos Post, Feb 14 2010

nonn

Cf. A001037, A058943 and A102569 for initial terms of underlying sequence. See also A058947, A011260, A059966, A000031 (n-bead necklaces but may have period dividing n).

a(n) = Sum_{i=0..n} A001037(i).

A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 2, 4, 6, 2, 4, 4, 2, 6, 4, 4, 8, 4, 2, 6, 2, 4, 8, 2, 4, 8, 4, 2, 6, 2, 2, 8, 14, 2, 6, 4, 8, 8, 4, 6, 6, 8, 12, 4, 4, 2, 8, 6, 2, 12, 8, 2, 8, 8, 2, 4, 4, 2, 4, 12, 6, 4, 6, 10, 20, 2, 4, 8, 2, 12, 6, 2, 2, 6, 4, 8, 16, 8, 2, 8, 4, 4, 16, 2

Offset: 0

Michael Schwartz, Aug 27 2020

a(n) > 1 for n > 0.
It appears that every term after a(2) is even.
It appears that a(2^n) is greater than each preceding term and is greater than or equal to each term up to a(2^(n+1)).
If a(n) = 2, then the nonzero shift register sequence is an m-sequence.

			For n = 3 = 11 in binary, the polynomial is 1+x+x^2 and the 2 shift register sequences are {00..., 01101...}.
For n = 4 = 100 in binary, the polynomial is 1+x^3 and the 4 shift register sequences are {000..., 001001..., 011011..., 111...}.
For n = 6 = 110 in binary, the polynomial is 1+x^2+x^3 and the 2 shift register sequences are {000..., 0010111001...}.
For n = 10 = 1010 in binary, the polynomial is 1+x^2+x^4 and the 4 shift register sequences are {0000..., 0001010001..., 0011110011..., 0110110...}.
For n = 11 = 1011 in binary, the polynomial in 1+x+x^2+x^4. Using a Fibonacci LSFR, if the current state of the register is 0001, the next input bit is 0+0+1=1, and the next state is 0011. If the current state is 0100, the next input bit is 0+0+0=0, and the next state is 1000. The 4 shift register sequences are {0000..., 00011010001..., 00101110010..., 1111...}.

a(2^n) = A000031(n+1).
A011260 counts how many 2's are in the interval [2^(n-1),(2^n)-1].
a(n) = 2 if and only if 2n+1 is in A091250.
Cf. A100447, A001037, A000016, A000013 (definition 2), A000020, A058947.
Cf. A011655..A011751 for examples of binary m-sequences.

cp's OEIS Frontend

A346019 Number of n X n invertible matrices over GF(2) that have order 2^n-1.

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A119512 Determinant of n X n matrix of first n^2 terms of A000020 number of primitive polynomials of degree n over GF(2).

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A173270 Partial sums of A001037, the number of degree-n irreducible polynomials over GF(2).

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A337442 Number of output sequences from the linear feedback shift register whose feedback polynomial coefficients (excluding the constant term) correspond to the binary representation of n.

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