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
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
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
KEYWORD
nonn,easy
AUTHOR
Tom Harris, Jun 12 2021
STATUS
approved