A352550 -id:A352550 - cp's OEIS frontend

A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.

1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0

Offset: 1

David Stanovsky, Oct 01 2014

A Mendelsohn triple system is affine if the associated quasigroup is affine, i.e, given by x*y=(1-f)(x)+f(y) over an abelian group (A,+) with an automorphism f.
For Steiner triple systems, the enumeration is settled by the following observation: a Steiner triple system is affine if and only if A=Z_3^n and f(x)=-x.
The existence spectrum (i.e., n such that a(n)>0) is A003136.
Comment from David Stanovsky, Mar 19 2022, added by N. J. A. Sloane, Mar 20 2022 (Start)
This is the sequence a(n) defined in the Donovan et al. paper.
The b(n) sequence defined there gives the number of non-affine systems.
The first 728 values of b(n) are now known: they are all zeros, except b(81) = 2, b(243) = 6, b(324) = 2, b(567)=4. We do not know b(729).
The reason is the following: it follows from the Galkin-Fischer-Smith theorem that, for n = m * 3^d, m not divisible by 3, we have b(n) = a(m) * b(3^d).
At the time of writing the paper, we could use known data about commutative Moufang loops to determine b(1) = b(3) = b(9) = b(27) = 0, and b(81) = 2. Later we managed to develop smarter enumeration methods that allowed us to determine b(243)=6: see Jedlička et al. (2007).
Since so many of the initial values of b(n), this does not have its own OEIS entry. (End)
Conjecture: This is the same sequences as A352561.(Note that A352561 has an explicit Dirichlet generating function.) - N. J. A. Sloane, Mar 21 2022

David Stanovsky, Table of n, a(n) for n = 1..1023
Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], 2014. [Published in Canad. Math. Bull. Vol. 59 (1), 2016 pp. 36-49.] See a(n) on page 9 of arXiv version.
Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, arXiv:1603.00608 [math.GR], 2016.
Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, Discrete Math. 340/3 (2017), 404--415.

Cf. A003136, A352550, A352561.

# For brevity, I do not exploit multiplicativity of a(n) here.
a := function(n)
    local count, gg, g, autg, conj, f, b, x;
    count := 0;
    for gg in AllGroups(Size, n, IsAbelian, true) do
        g := Image(IsomorphismPermGroup(gg), gg);
        autg := AutomorphismGroup(g);
        conj := List(ConjugacyClasses(autg), x->Representative(x));
        for f in conj do
            b := true;
            for x in g do
                if not
                   Image(f, Image(f, x))*Image(f, x^-1)*x = ()
                then b := false; break;
                fi;
            od;
            if b then count := count + 1; fi;
        od;
    od;
    return count;
end;

A038540 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 40.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 0, 3, 5, 1, 0, 4, 2, 0, 2, 5, 0, 5, 0, 2, 0, 0, 0, 6, 2, 2, 10, 0, 0, 2, 2, 7, 0, 0, 0, 10, 2, 0, 4, 3, 2, 0, 2, 0, 5, 0, 0, 10, 1, 2, 0, 4, 2, 10, 0, 0, 0, 0, 0, 4, 0, 2, 0, 11, 2, 0, 2, 0, 0, 0, 2, 15, 0, 2, 4, 0, 0, 4, 2, 5, 20, 2, 2, 0, 0, 2, 0, 0, 2, 5, 0, 0, 4, 0, 0, 14

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

Equivalently, number of modules with n elements over ring Z[sqrt(10)].

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

Edited by M. F. Hasler, Feb 18 2022

Revised by N. J. A. Sloane, Mar 21 2022

A038541 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -20.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 3, 5, 1, 0, 4, 0, 2, 2, 5, 0, 5, 0, 2, 4, 0, 2, 6, 2, 0, 10, 4, 2, 2, 0, 7, 0, 0, 2, 10, 0, 0, 0, 3, 2, 4, 2, 0, 5, 2, 2, 10, 5, 2, 0, 0, 0, 10, 0, 6, 0, 2, 0, 4, 2, 0, 10, 11, 0, 0, 2, 0, 4, 2, 0, 15, 0, 0, 4, 0, 0, 0, 0, 5, 20, 2, 2, 8, 0, 2, 4, 0, 2, 5, 0, 4, 0, 2, 0, 14

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

Equivalently, number of modules with n elements over the ring Z[sqrt(-5)].

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

Edited by M. F. Hasler, Feb 18 2022

Revised by N. J. A. Sloane, Mar 21 2022

A352567 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -19.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 4, 2, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352566.

A352551 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 8.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 3, 1, 0, 0, 0, 0, 2, 0, 5, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 4, 0, 0, 2, 7, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 5, 1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 2, 11, 0, 0, 0, 4, 0, 0, 2, 3, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352561 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -3.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Appears to be the same sequence as A248107.

Cf. A038540, A038541, A248107, A352550-A352567.

A352566 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -15.

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 0, 10, 2, 2, 0, 5, 0, 0, 1, 20, 2, 4, 2, 5, 0, 0, 2, 10, 2, 0, 3, 0, 0, 2, 2, 36, 0, 4, 0, 10, 0, 4, 0, 10, 0, 0, 0, 0, 2, 4, 2, 20, 1, 4, 2, 0, 2, 6, 0, 0, 2, 0, 0, 5, 2, 4, 0, 65, 0, 0, 0, 10, 2, 0, 0, 20, 0, 0, 2, 10, 0, 0, 2, 20, 5, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 10, 2, 4, 2, 36, 0, 2, 0, 10, 0, 4, 0, 0, 0, 4, 2, 15, 2, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352552 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 12.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 0, 3, 2, 0, 2, 2, 2, 0, 0, 5, 0, 2, 0, 0, 0, 2, 2, 3, 1, 2, 3, 0, 0, 0, 0, 7, 2, 0, 0, 4, 2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 2, 5, 1, 1, 0, 4, 0, 3, 0, 0, 0, 0, 2, 0, 2, 0, 0, 11, 0, 2, 0, 0, 2, 0, 2, 6, 2, 2, 1, 0, 0, 2, 0, 0, 5, 0, 2, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352553 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 13.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 0, 0, 5, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 10, 0, 2, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 10, 0, 0, 0, 0, 2, 0, 0, 2

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352554 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 17.

Original entry on oeis.org

1, 2, 0, 5, 0, 0, 0, 10, 1, 0, 0, 0, 2, 0, 0, 20, 1, 2, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 36, 0, 2, 0, 5, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 10, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 65, 0, 0, 2, 5, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 5, 2, 0, 2, 20, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 1, 0, 0, 0, 0, 0, 2, 110, 0, 0, 0, 0, 0, 4, 0, 10, 2, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

cp's OEIS Frontend

A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

A038540 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 40.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A038541 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -20.

Views

Author

Keywords

Comments

Crossrefs

Extensions

A352567 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -19.

Views

Author

Keywords

Comments

Crossrefs

A352551 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 8.

Views

Author

Keywords

Comments

Crossrefs

A352561 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -3.

Views

Author

Keywords

Comments

Crossrefs

A352566 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -15.

Views

Author

Keywords

Comments

Crossrefs

A352552 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 12.

Views

Author

Keywords

Comments

Crossrefs

A352553 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 13.

Views

Author

Keywords

Comments

Crossrefs

A352554 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 17.

Views

Author

Keywords

Comments

Crossrefs