OFFSET
0,2
COMMENTS
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, Mar 25 2002
(-1)^n*B_{2n}/(-4n) = Integral_{t>=0} t^(2n-1)/(exp(2*Pi*t) - 1)dt. - Benoit Cloitre, Apr 04 2002
Michael Lugo (see link) conjectures, and Peter McNamara proves, that a(n) = gcd_{ primes p > 2n+1 } (p^(2n) - 1). - Tanya Khovanova, Feb 21 2009 [edited by Charles R Greathouse IV, Dec 03 2014]
REFERENCES
Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 220.
F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 286.
Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.)
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
P. J. Cameron and D. A. Preece, Notes on primitive lambda-roots
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points, arXiv:math/0204173 [math.NT], 2002.
Michael Lugo, A little number theory problem (2008)
Eric Weisstein's World of Mathematics, Eisenstein Series.
FORMULA
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.
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, Mar 25 2002
MAPLE
1, seq(denom(bernoulli(2*n)/(-4*n)), n=1 .. 100); # Robert Israel, Dec 03 2014
MATHEMATICA
a[n_] := Denominator[BernoulliB[2n]/(-4n)]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 20 2011 *)
PROG
(PARI) a(n) = if (n == 0, 1, denominator(bernfrac(2*n)/(-4*n))); \\ Michel Marcus, Sep 10 2013
(Magma) [1] cat [Denominator(Bernoulli(2*n)/(-4*n)):n in [1..35]]; // G. C. Greubel, Sep 19 2019
(Sage) [1]+[denominator(bernoulli(2*n)/(-4*n)) for n in (1..35)] # G. C. Greubel, Sep 19 2019
(GAP) Concatenation([1], List([1..35], n-> DenominatorRat(Bernoulli(2*n)/(-4*n)) )); # G. C. Greubel, Sep 19 2019
CROSSREFS
KEYWORD
nonn,easy,frac,nice
AUTHOR
EXTENSIONS
Thanks to Michael Somos for helpful comments.
STATUS
approved