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

Author

Tom Harris, Jun 12 2021

Keywords

• nonn
• easy

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

• 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.

Crossrefs

Cf. A006863, A079612. Divides A048648.

Programs

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_]

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.)