A028362 -id:A028362 - cp's OEIS frontend

A203478 a(n) = v(n+1)/v(n), where v = A203477.

3, 30, 1080, 146880, 77552640, 161309491200, 1331771159347200, 43809944057885491200, 5753472333233985788313600, 3019422280481195741706977280000, 6335279362770913356551778761441280000

Offset: 1

Clark Kimberling, Jan 02 2012

G. C. Greubel, Table of n, a(n) for n = 1..55

Cf. A028362, A093883, A164051, A203307, A203477.

[(&*[2^j + 2^n: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 28 2023

(* First program *)
f[j_]:= 2^(j-1); z = 13;
v[n_]:= Product[Product[f[k] + f[j], {j,k-1}], {k,2,n}]
Table[v[n], {n,z}]                       (* A203477 *)
Table[v[n+1]/v[n], {n,z-1}]              (* A203478 *)
Table[v[n]*v[n+2]/(2*v[n+1]^2), {n,22}]  (* A164051 *)
(* Second program *)
Table[Product[2^j +2^n, {j,0,n-1}], {n,20}] (* G. C. Greubel, Aug 28 2023 *)

a(n)=prod(i=0,n-1,2^i+2^n) \\ Charles R Greathouse IV, Feb 16 2021

[product(2^j + 2^n for j in range(n)) for n in range(1,21)] # G. C. Greubel, Aug 28 2023

a(n) = A028362(n+1) * 2^(n*(n-1)/2). - Charles R Greathouse IV, Feb 16 2021

a(n) = Product_{j=0..n-1} (2^j + 2^n). - G. C. Greubel, Aug 28 2023

A277355 a(n) = Product_{k=1..n} (2^k + k).

Original entry on oeis.org

1, 3, 18, 198, 3960, 146520, 10256400, 1384614000, 365538096000, 190445348016000, 196920489848544000, 405459288598152096000, 1665626757561208810368000, 13666467545789718289069440000, 224102734815859800504160677120000, 7346759955468331839927899478024960000

Offset: 0

Vaclav Kotesovec, Oct 10 2016

nonn

Cf. A005329, A028362.

Table[Product[2^k+k, {k, 1, n}], {n, 0, 15}]

a(n) ~ c * 2^(n*(n+1)/2), where c = Product_{k>=1} (1 + k/2^k) = 5.52995584900...

A119600 a(n) = 4*Product_{i=1..n-1} (3^i+1)^2.

Original entry on oeis.org

4, 4, 64, 6400, 5017600, 33738342400, 2008645953126400, 1070407428421058560000, 5124408580006984170864640000, 220656234047362257307900743516160000, 85495432669493277396354169745064287272960000, 298114237913837782686540845369489025952802406400000000, 9355246290649672947599943358541996936410690283965618585600000000

Offset: 0

N. J. A. Sloane, Jun 04 2006

nonn

R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des Nombres de Bordeaux, 14 (2002), 73-85.

Cf. A028362, A290000.

Table[4*Product[(3^i+1)^2,{i,n-1}],{n,0,12}] (* James C. McMahon, Sep 17 2024 *)

a(n) = 4*prod(i=1, n-1, (3^i+1)^2); \\ Michel Marcus, Oct 28 2015

a(n) = 4*A290000(n)^2. - Vaclav Kotesovec, Sep 17 2024

A155104 Numbers appearing in the fourth column of A155103.

Original entry on oeis.org

1, 5, 45, 765, 25245, 1640925, 211679325, 54401586525, 27908013887325, 28605714234508125, 58613108466507148125, 240137905387279785868125, 1967449858837983285617548125, 32236665937060356134843526028125

Offset: 1

Mats Granvik, Jan 20 2009

nonn

			a(1)=1, a(2)=1*5, a(3)=1*5*9, a(4)=1*5*9*17, a(5)=1*5*9*17*33, a(6)=1*5*9*17*33*65 and so on. See A000051 for continuation.

Cf. A000051, A155105.

Table[QPochhammer[-4, 2, n - 1], {n, 14}] (* Arkadiusz Wesolowski, Oct 29 2012 *)

a(n) = prod(k=2..n) (1+2^k), n>1. - Mats Granvik, Simone Severini, Dec 04 2010

a(n) = A028362(n+1)/3. - Vladimir Reshetnikov, Sep 16 2016

More terms from Mats Granvik, Dec 05 2010

A219732 a(n) = (Product_{i=1..n-1} (2^i + 1)) modulo (2^n - 1).

Original entry on oeis.org

0, 0, 1, 0, 1, 9, 1, 0, 74, 33, 1, 1170, 1, 129, 15101, 0, 1, 187758, 1, 67650, 615700, 2049, 1, 4793490, 3247204, 8193, 262658, 4227330, 1, 480000312, 1, 0, 2458463380, 131073, 10787055277, 19903096980, 1, 524289, 67117058, 567489872400, 1, 2686322969514, 1

Offset: 1

Michel Marcus, Nov 26 2012

nonn

E. Vantieghem proved that a(n) = 1 if and only if n is an odd prime. - Michel Marcus, Nov 26 2012

E. Vantieghem, On a congruence only holding for primes II, arXiv:0812.2841 [math.NT], 2008-2009.

Cf. A028362.

Join[{0}, Table[m = 2^n - 1; prod = 1; Do[prod = Mod[prod*(2^i + 1), m], {i, n - 1}]; prod, {n, 2, 40}]] (* T. D. Noe, Nov 27 2012 *)

a(m) = {for (n=1, m, print1(prod(j=1, n-1, 2^j+1) % (2^n - 1), ", "););}

a(n)=if(n>2,my(m=2^n-1);lift(prod(i=1,n-1,Mod(2,m)^i+1)),0) \\ Charles R Greathouse IV, Nov 26 2012

a(n) = A028362(n) modulo (2^n - 1).

A323357 Number of binary self-dual codes of length 2n (up to permutation equivalence) that have a unique automorphism group size.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 9, 16, 23, 42, 68, 94, 124, 159, 187, 212

Offset: 1

Nathan J. Russell, Jan 12 2019

Two codes are said to be permutation equivalent if permuting the columns of one code results in the other code.
If permuting the columns of a code results in the same identical code the permutation is called an automorphism.
The automorphisms of a code form a group called the automorphism group.
Some codes have automorphism groups that contain the same number of elements.  There are situations, both trivial and otherwise, that codes of different lengths can have the same size automorphism groups.
Some codes have automorphism group sizes that are unique to the code. This sequence only compares automorphism group sizes for codes with the same length.

			There are a(18) = 212 binary self-dual codes (up to permutation equivalence) of length 2*18 = 36 that have a unique automorphism group size.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.

For self-dual codes see A028362, A003179, A106162, A028363, A106163, A269455, A120373; for automorphism groups see A322299, A322339.

A353196 Number of stabilizer states on n qubits.

Original entry on oeis.org

6, 60, 1080, 36720, 2423520, 315057600, 81284860800, 41780418451200, 42866709330931200, 87876754128408960000, 360118938418219918080000, 2950814581398894008747520000, 48352047730802277227336862720000, 1584496604138390624739828991334400000

Offset: 1

James Rayman, Apr 29 2022

A stabilizer state is a quantum state on n qubits prepared by applying a series of Hadamard, CNOT, and S gates to the all-zero state. There are only a finite number of such states for any n.

			For n = 1, the a(1) = 6 states are |0>, |1>, |+>, |->, |i>, and |-i>.

D. Gross, Hudson's Theorem for finite-dimensional quantum systems, arXiv:quant-ph/0602001, 2006-2007.

Cf. A000079, A003956, A028362.

Table[2^n * QPochhammer[-2, 2, n], {n, 13}] (* Amiram Eldar, Aug 17 2025 *)

def a(n):
    ans = 2 ** n
    for i in range(1, n+1):
        ans *= 2 ** i + 1
    return ans

from math import prod
def A353196(n): return prod((1<Chai Wah Wu, Jun 20 2022

a(n) = 2^n*Product_{i=1..n} (2^i+1).
a(n) = A000079(n)*A028362(n+1).
a(n) ~ c * 2^(n*(n+3)/2) where c = Product_{k>=1} (1 + 1/2^k) = A079555. - Amiram Eldar, Aug 17 2025

A079674 Continued fraction expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 14, 1, 3, 1, 1, 6, 9, 18, 7, 1, 27, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 4, 7, 4, 36, 1, 6, 1, 28, 1, 1, 6, 1, 3, 2, 150, 2, 1, 1, 36, 3, 2, 6, 4, 1, 2, 1, 1, 9, 1, 12, 12, 6, 7, 7, 2, 4, 3, 56, 1, 22, 1, 7, 2, 1, 1, 36, 4, 1, 3, 1, 1, 2, 1, 10, 1, 1, 82, 16, 2, 1, 1, 6, 15, 1, 2, 1, 5, 1, 1

Offset: 0

Benoit Cloitre, Jan 25 2003

Cf. A028362.

Cf. A079555 (decimal expansion).

ContinuedFraction[QPochhammer[-1/2,1/2],97] (* Stefano Spezia, Aug 19 2025 *)

Offset changed by Andrew Howroyd, Jul 06 2024

A321946 Number of divisors for the automorphism group size having the largest number of divisors for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 4, 10, 28, 36, 66, 144, 192, 340, 570, 1200, 1656, 3456, 5616, 9072, 10752, 22176

Offset: 1

Nathan J. Russell, Dec 12 2018

A code is usually represented by a generating matrix. The row space of the generating matrix is the code itself.
Self-dual codes are codes such all codewords are pairwise orthogonal to each other.
Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.
The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.
The values in the sequence are not calculated lower bounds. For each n there exists a binary self-dual code of length 2n with an automorphism group of size a(n).
Binary self-dual codes have been classified (accounted for) up to a certain length. The classification process requires the automorphism group size be known for each code. There is a mass formula to calculate the number of distinct binary self-dual codes of a given length. Sequence A028362gives this count. The automorphism group size allows researchers to calculate the number of codes that are permutationally equivalent to a code. Each new binary self-dual code C of length m that is discovered will account for m!/aut(C) codes in the total number calculated by the mass formula. Aut(C) represents the automorphism size of the code C. Sequence A003179 gives number of binary self-dual codes up to permutation equivalence.
The values in the sequence are not calculated by a formula or algorithm.  They are the result of calculating the number of divisors for every automorphism group of every binary self-dual code.
The number of divisors a(n) does count 1 and the number itself.
In general the automorphism group size with the largest number of divisors is not unique.
In general the automorphism group size with the largest number of divisors is not the largest group automorphism group size for a given binary self-dual code length.

			There is one binary self-dual code of length 2*14=28 having an automorphism group size of 1428329123020800.  This number has a(14) = 5616 divisors (including 1 and 1428329123020800).  The automorphism size of 1428329123020800 represents the automorphism size with the largest number of divisors for a binary self-dual code of length 2*14=28.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.

Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163, A269455, A120373.

Cf. Self-Dual Code Automorphism Groups A322299, A322339.

A322309 Largest automorphism group size for a binary self-dual code of length 2n.

Original entry on oeis.org

2, 8, 48, 1344, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000, 1371195958099968000, 46620662575398912000

Offset: 1

Nathan J. Russell, Dec 03 2018

nonn

A code is usually represented by a generating matrix.  The row space of the generating matrix is the code itself.
Self-dual codes are codes such that all codewords of the code are pairwise orthogonal to each other.
Two codes are called permutation equivalent if one code can be obtained by permuting the coordinates (columns) of the other code.
The automorphism group of a code is the set of permutations of the coordinates (columns) that result in the same identical code.
The values in the sequence are not calculated upper bounds.  For each n there exists a binary self-dual code of length 2n with an automorphism group of size a(n).
Binary self-dual codes have been classified (accounted for) up to a certain length. The classification process requires the automorphism group size be known for each code.  There is a mass formula to calculate the number of distinct binary self-dual codes of a given length.  The automorphism group size allows researchers to calculate the number of codes that are permutationally equivalent to a code.  Each new binary self-dual code C of length m that is discovered will account for m!/aut(C) codes in the total number calculated by the mass formula. Aut(C) represents the automorphism size of the code C.

			The largest automorphism group size a binary self-dual code of length 2*16=32 is a(16) = 1371195958099968000.

W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, 2003, Pages 338-393.

Cf. Self-Dual Codes A028362, A003179, A106162, A028363, A106163.

cp's OEIS Frontend

A203478 a(n) = v(n+1)/v(n), where v = A203477.

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A277355 a(n) = Product_{k=1..n} (2^k + k).

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A119600 a(n) = 4*Product_{i=1..n-1} (3^i+1)^2.

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A155104 Numbers appearing in the fourth column of A155103.

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A219732 a(n) = (Product_{i=1..n-1} (2^i + 1)) modulo (2^n - 1).

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A323357 Number of binary self-dual codes of length 2n (up to permutation equivalence) that have a unique automorphism group size.

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A353196 Number of stabilizer states on n qubits.

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A079674 Continued fraction expansion of Product_{k>=1} (1 + 1/2^k) = 2.384231029031371...

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