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User: Tom Harris

Tom Harris has authored 5 sequences.

A371022 Numbers k with the property that there is a finite set W of groups of order divisible by k such that if k divides the order of a group G, then G has a subgroup isomorphic to a group in W.

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 34, 37, 41, 43, 47, 49, 50, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199

Offset: 1

Tom Harris, Mar 08 2024

These are called Cauchy numbers in Cameron et al., where they are proved to be the following set: 6 U prime powers U numbers of the form 2*p^a where p is a Fermat prime greater than 3.

Peter J. Cameron, David Craven, Hamid Reza Dorbidi, Scott Harper, and Benjamin Sambale, Minimal Cover Groups, arXiv:2311.15652 [math.GR], 2023-2024.

Cf. A000961, A019434.

A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

Original entry on oeis.org

1, 2, 2, 48, 1, 1, 1, 240, 1, 2, 2, 1008, 1, 1, 1, 480, 1, 2, 2, 528, 1, 1, 691, 65520, 1, 2, 2, 48, 1, 1, 3617, 16320, 1, 2, 87734, 57456, 1, 1, 174611, 13200, 1, 2, 155366, 1104, 1, 1, 236364091, 131040, 1, 2, 1315862, 48, 1, 1, 3392780147, 6960, 1, 2

Offset: 0

Tom Harris, Jun 12 2021

a(n) is known for n != 0 mod 4 and is related to the Bernoulli numbers via the Riemann zeta function. See Section VI.9 of Weibel's K-book.
K_0(Z) = Z is classical, so a(0) = 1. Rognes proved that K_4(Z) = 0 in 2000 so a(4) = 1.
Otherwise the value of a(4i) = 1 is conjectural. Kurihara observed that this follows from the Kummer-Vandiver conjecture (and in fact is equivalent with it). The Kummer-Vandiver conjecture has been verified for primes up to 163 million, from which it follows that a(4i) must be at least that large if it is not 1.

C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compositio Math. 81 (1992), 223-236.
J. Rognes, K_4(Z) is the trivial group, Topology 30 (2000), 267-281.
C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

Cf. A345225 (the order of the 2-primary subgroup, divides a(n)).

A000367 / A006863 (numerator / denominator of B_2n/4n, where B_m are the Bernoulli numbers).

from sympy import bernoulli
def a(n):
    n_ = n % 8
    k = n//4 + 1
    b = bernoulli(2*k)/(4*k)
    d = {0:1, 1:2, 4:1, 5:1}
    if n_ == 2:
        return 2 * b.numerator
    elif n_ == 3:
        return 2 * b.denominator
    elif n_ == 6:
        return -1 * b.numerator
    elif n_ == 7:
        return b.denominator
    else:
        return d[n_]

Let k be the integer part of 1 + n/4.
a(n) is:
              1 if n = 0 mod 8
              2 if n = 1 mod 8
   2*A000367(k) if n = 2 mod 8
   2*A006863(k) if n = 3 mod 8
              1 if n = 4 mod 8
              1 if n = 5 mod 8
  -1*A000367(k) if n = 6 mod 8
     A006863(k) if n = 7 mod 8.

A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

Original entry on oeis.org

1, 2, 1, 24, 1, 1, 1, 240, 2, 2, 1, 504, 1, 1, 1, 480, 2, 2, 1, 264, 1, 1, 1, 65520, 2, 2, 1, 24, 1, 1, 1, 16320, 2, 2, 1, 28728, 1, 1, 1, 13200, 2, 2, 1, 552, 1, 1, 1, 131040, 2, 2, 1, 24, 1, 1, 1, 6960, 2, 2, 1, 171864, 1, 1, 1, 32640, 2, 2, 1, 24, 1, 1, 1

Offset: 0

Tom Harris, Jun 12 2021

Im(J) is a finite cyclic subgroup of Pi_n^S and has known order a(n) calculated by Adams using the Adams conjecture, subsequently proven by Quillen. When n is 3 or 7 mod 8 the value a(n) is related to the Bernoulli numbers; the other values of a(n) are 8-periodic (after an exceptional n=0).

D. Ravenel, Complex cobordism and stable homotopy groups of spheres (2ed), AMS Chelsea Publishing, (2003), ISBN: 978-0-8218-2967-7.

J.F. Adams, On the groups J(x), IV, Topology 5 (1966), 21-71.
D.G. Quillen, The Adams conjecture, Topology 10 (1971), 1-10.

Cf. A006863, A079612. Divides A048648.

from sympy import bernoulli
def a(n):
    if n == 0:
        return 1
    n_ = n % 8
    d = {0:2, 1:2, 2:1, 4:1, 5:1, 6:1}
    if n_ in [3, 7]:
        k = (n+1)//4
        return (bernoulli(2*k)/(4*k)).denominator
    else:
        return d[n_]

a(n) is:
                 2 if n = 0 or 1 mod 8 (except a(0) = 1)
                 1 if n = 2, 4, 5 or 6 mod 8
  A006863((n+1)/4) if n = 3 or 7 mod 8.
(A006863(k) = denominator of B_2k/4k, where B_m are the Bernoulli numbers.)

A345225 a(n) = order of 2-primary subgroup of the group K_n(Z).

Original entry on oeis.org

1, 2, 2, 16, 1, 1, 1, 16, 1, 2, 2, 16, 1, 1, 1, 32, 1, 2, 2, 16, 1, 1, 1, 48, 1, 2, 2, 16, 1, 1, 1, 64, 1, 2, 2, 16, 1, 1, 1, 80, 1, 2, 2, 16, 1, 1, 1, 96, 1, 2, 2, 16, 1, 1, 1, 112, 1, 2, 2, 16, 1, 1, 1, 128, 1, 2, 2, 16, 1, 1, 1, 144, 1, 2, 2, 16, 1, 1, 1, 160

Offset: 0

Tom Harris, Jun 11 2021

The algebraic K-theory groups of the integers are not yet (as of June 2021) completely known, however the 2-primary part is settled.

In fact, the 2-primary part of the abelian group K_n(Z) is the cyclic group Z/a(n)Z.

C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-9132-.

Tom Harris, Table of n, a(n) for n = 0..9999
C. Weibel, The 2-torsion in the K-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615-620.

def a(n):
    n_ = (n % 8)
    d = {0:1, 1:2, 2:2, 3:16, 4:1, 5:1, 6:1}
    if n_ == 7:
        return 2*(n+1)
    else:
        return d[n_]

a(n) is:
        1 if n mod 8 = 0, 4, 5, or 6
        2 if n mod 8 = 1 or 2
       16 if n mod 8 = 3
  2*(n+1) if n mod 8 = 7.
(The main result of Weibel's 1997 paper on the 2-torsion in the K-theory of the integers; Corollary 9.8 of Weibel's K-book.)

A345209 Number of Petrie polygons on the regular triangular map corresponding to the principal congruence subgroup Gamma(n) of the modular group.

Original entry on oeis.org

1, 1, 3, 4, 6, 6, 21, 16, 27, 12, 66, 24, 78, 42, 36, 64, 136, 162, 190, 48, 252, 132, 253, 192, 150, 156, 243, 168, 870, 72, 496, 256, 396, 816, 252, 648, 666, 1140, 468, 384, 1722, 504, 903, 1056, 324, 1518, 3243, 1536, 1029, 300, 816, 624, 1378, 1458, 3960, 1344, 1140, 1740, 1770, 576

Offset: 1

Tom Harris, Jun 10 2021

To each principal congruence subgroup Gamma(n) of the modular group Gamma = PSL(2,Z) there corresponds a regular triangular map (it is the quotient of the Farey map by Gamma(n)). A Petrie polygon is a closed left-right zig-zagging path on the map. a(n) is the number of such paths.

			The regular triangular map corresponding to Gamma(3) is the tetrahedron; one can easily check by hand that there are 3 distinct closed left-right zigzag paths (Petrie polygons) along the edges of the tetrahedron, so a(3) = 3.
Similarly, there are a(4) = 4 and a(5) = 6 such paths on the octahedron and the icosahedron, the maps corresponding to Gamma(4), and Gamma(5) respectively.
The map corresponding to Gamma(7) is the Klein map on his quartic curve. There are 21 Petrie polygons on this map; Klein drew 3 of them in his 1878 paper on the quartic, and the others can be found by rotating these through 2*Pi*k/7, k=1,...,6.

Tom Harris, Table of n, a(n) for n = 1..1000
F. Klein, Ueber die Transformation siebenter Ordnungder elliptischen Funktionen, Mathematische Annalen, 14 (1878), 428-471.
D. Singerman and J. Strudwick, Petrie polygons, Fibonacci sequences and Farey maps, Ars Mathematica Contemporanea, 10 (2016), 349-357.

A301759 gives the lengths of the Petrie polygons on the map in question.

b[n_] := (n^3/2) Times @@ (1-1/Select[Range[n], Mod[n, #] == 0 && PrimeQ[#]&]^2);
c[n_] := With[{F = Fibonacci}, For[k = 1, True, k++, If[Mod[F[k], n] == 0 && (Mod[F[k+1], n] == 1 || Mod[F[k+1], n] == n-1), Return[k]]]];
a[n_] := If[n<3, 1, b[n]/c[n]];
Array[a, 60] (* Jean-François Alcover, Jun 11 2021 *)
Table[((n^3/2^Boole[n > 1]) Product[1 - 1/k^2, {k, Select[Divisors[n], PrimeQ]}])/NestWhile[# + 1 &, 1, ! (Mod[Fibonacci[#], n] == 0 && With[{f = Mod[Fibonacci[# + 1], n]}, f == 1 || f == n - 1]) &], {n, 60}] (* Jan Mangaldan, Sep 12 2021 *)

from sympy import primefactors
def a(n):
    # degenerate cases
    if n == 1 or n == 2:
        return 1
    # calculate index of Γ(n) in Γ
    index = n**3
    for p in primefactors(n):
        index *= (p**2 - 1)
        index //= p**2
    index //= 2
    # calculate pisano semiperiod
    sigma = 1
    a, b = 1, 1
    while (a,b) != (0,1) and (a,b) != (0, n - 1):
        a, b = b, (a + b) % n
        sigma += 1
    # number of petrie polygons = index / sigma
    return index // sigma

a(n) = A001766(n)/A301759(n), n >= 3 (Corollary 7.3 of Singerman & Strudwick)

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User: Tom Harris

A371022 Numbers k with the property that there is a finite set W of groups of order divisible by k such that if k divides the order of a group G, then G has a subgroup isomorphic to a group in W.

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A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

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A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.

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A345225 a(n) = order of 2-primary subgroup of the group K_n(Z).

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A345209 Number of Petrie polygons on the regular triangular map corresponding to the principal congruence subgroup Gamma(n) of the modular group.

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