%I A006863 M5150
%S A006863 1,24,240,504,480,264,65520,24,16320,28728,13200,552,131040,
%T A006863 24,6960,171864,32640,24,138181680,24,1082400,151704,5520,
%U A006863 1128,4455360,264,12720,86184,13920,1416,6814407600,24
%N A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.
%C A006863 Carmichael defines lambda(n) to be the exponent of the group U(n) of 
               units of the integers mod n. He shows that given m there is a number 
               lambda^*(m) such that lambda(n) divides m if and only if n divides 
               lambda^*(m). He gives a formula for lambda^*(m), equivalent to the 
               one I've quoted for even m. (We have lambda^*(m)=2 for any odd m.) 
               The present sequence gives the values of lambda^*(2m) for positive 
               integers m. - Peter J. Cameron (p.j.cameron(AT)qmul.ac.uk), Mar 25 
               2002
%C A006863 (-1)^n*B_{2n}/(-4n) = integral(t=O,infinity,t^(2n-1)/(exp(2Pi*t)-1)dt) 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 04 2002
%C A006863 Comment from Tanya Khovanova, Feb 21 2009: According to the godplaysdice.blogspot.com 
               link given below, a(n) = GCD_{ primes p > 2n+1 } (p^(2n) - 1).
%D A006863 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006863 Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals 
               and Asymptotic Expansions, p. 220.
%D A006863 R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. 
               Soc. 16 (1909-10), 232-238.
%D A006863 F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, 
               p. 130.
%D A006863 J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, 
               p. 286.
%D A006863 Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture 
               Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, 
               pp. 123-133.
%D A006863 R. C. Vaughan and T. D. Wooley, Waring's problem: a survey, pp. 285-324 
               of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett 
               et al., Peters, 2003. (The function K(2n), see p. 303.)
%H A006863 T. D. Noe, <a href="b006863.txt">Table of n, a(n) for n=0..10000</a>
%H A006863 G. Everest, Y. Puri and T. Ward, <a href="http://arXiv.org/abs/math.NT/
               0204173">Integer sequences counting periodic points</a>
%H A006863 Michael Lugo, <a href="http://godplaysdice.blogspot.com/2008/05/little-number-theory-problem.html">
               A little number theory problem</a> [From Tanya Khovanova, Feb 21 
               2009]
%H A006863 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               EisensteinSeries.html">Link to a section of The World of Mathematics.</
               a>
%H A006863 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%F A006863 B_{2k}/(4k) = -1/2*\zeta(1-2k). For n>0, a(n) = gcd k^L (k^{2n}-1) where 
               k ranges over all the integers and L is as large as necessary.
%F A006863 Product of 2^{a+2} (where 2^a exactly divides 2*n) and p^{a+1} (where 
               p is an odd prime such that p-1 divides 2*n and p^a exactly divides 
               2*n). - Peter J. Cameron (p.j.cameron(AT)qmul.ac.uk), Mar 25 2002
%Y A006863 Numerators are A001067. Cf. A000367/A002445, A002322, A079612.
%Y A006863 Sequence in context: A003264 A003272 A003245 this_sequence A052663 A167548 
               A052796
%Y A006863 Adjacent sequences: A006860 A006861 A006862 this_sequence A006864 A006865 
               A006866
%K A006863 nonn,easy,frac,nice
%O A006863 0,2
%A A006863 N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit, Simon Plouffe 
               (simon.plouffe(AT)gmail.com)
%E A006863 Thanks to Michael Somos for helpful comments.