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

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

0,2

Comments

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).

References

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

Links

Table of n, a(n) for n=0..70.

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.

Formula

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.)

Prog

(Python)
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_]

Crossrefs

Cf. A006863, A079612. Divides A048648.

Sequence in context: A059360 A368016 A279308 * A108778 A271530 A062763

Adjacent sequences: A345259 A345260 A345261 * A345263 A345264 A345265

Keyword

nonn,easy

Author

Tom Harris, Jun 12 2021

Status

approved