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

%I A345262 #11 Jun 27 2025 15:30:04
%S A345262 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,
%T A345262 24,1,1,1,16320,2,2,1,28728,1,1,1,13200,2,2,1,552,1,1,1,131040,2,2,1,
%U A345262 24,1,1,1,6960,2,2,1,171864,1,1,1,32640,2,2,1,24,1,1,1
%N A345262 a(n) is the order of the image of the J-homomorphism in the stable homotopy groups of spheres.
%C A345262 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 A345262 D. Ravenel, Complex cobordism and stable homotopy groups of spheres (2ed), AMS Chelsea Publishing, (2003), ISBN: 978-0-8218-2967-7.
%H A345262 J.F. Adams, <a href="https://doi.org/10.1016/0040-9383(66)90004-8">On the groups J(x), IV</a>, Topology 5 (1966), 21-71.
%H A345262 D.G. Quillen, <a href="https://doi.org/10.1016/0040-9383(71)90018-8">The Adams conjecture</a>, Topology 10 (1971), 1-10.
%F A345262 a(n) is:
%F A345262                  2 if n = 0 or 1 mod 8 (except a(0) = 1)
%F A345262                  1 if n = 2, 4, 5 or 6 mod 8
%F A345262   A006863((n+1)/4) if n = 3 or 7 mod 8.
%F A345262 (A006863(k) = denominator of B_2k/4k, where B_m are the Bernoulli numbers.)
%o A345262 (Python)
%o A345262 from sympy import bernoulli
%o A345262 def a(n):
%o A345262     if n == 0:
%o A345262         return 1
%o A345262     n_ = n % 8
%o A345262     d = {0:2, 1:2, 2:1, 4:1, 5:1, 6:1}
%o A345262     if n_ in [3, 7]:
%o A345262         k = (n+1)//4
%o A345262         return (bernoulli(2*k)/(4*k)).denominator
%o A345262     else:
%o A345262         return d[n_]
%Y A345262 Cf. A006863, A079612. Divides A048648.
%K A345262 nonn,easy
%O A345262 0,2
%A A345262 _Tom Harris_, Jun 12 2021