%I M5150 #65 Sep 08 2022 08:44:35

%S 1,24,240,504,480,264,65520,24,16320,28728,13200,552,131040,24,6960,

%T 171864,32640,24,138181680,24,1082400,151704,5520,1128,4455360,264,

%U 12720,86184,13920,1416,6814407600,24

%N Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

%C 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

%C (-1)^n*B_{2n}/(-4n) = Integral_{t>=0} t^(2n-1)/(exp(2*Pi*t) - 1)dt. - _Benoit Cloitre_, Apr 04 2002

%C 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]

%D Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 220.

%D F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 130.

%D J. W. Milnor and J. D. Stasheff, Characteristic Classes, Princeton, 1974, p. 286.

%D Douglas C. Ravenel, Complex cobordism theory for number theorists, Lecture Notes in Mathematics, 1326, Springer-Verlag, Berlin-New York, 1988, pp. 123-133.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D 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 T. D. Noe, <a href="/A006863/b006863.txt">Table of n, a(n) for n = 0..10000</a>

%H P. J. Cameron and D. A. Preece, <a href="http://www.maths.qmul.ac.uk/~pjc/csgnotes/lambda.pdf">Notes on primitive lambda-roots</a>

%H R. D. Carmichael, <a href="http://dx.doi.org/10.1090/S0002-9904-1910-01892-9">Note on a new number theory function</a>, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.

%H G. Everest, Y. Puri and T. Ward, <a href="https://arxiv.org/abs/math/0204173">Integer sequences counting periodic points</a>, arXiv:math/0204173 [math.NT], 2002.

%H Michael Lugo, <a href="http://godplaysdice.blogspot.com/2008/05/little-number-theory-problem.html">A little number theory problem</a> (2008)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series.</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F 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 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

%p 1,seq(denom(bernoulli(2*n)/(-4*n)), n=1 .. 100); # _Robert Israel_, Dec 03 2014

%t a[n_] := Denominator[BernoulliB[2n]/(-4n)]; Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Mar 20 2011 *)

%o (PARI) a(n) = if (n == 0, 1, denominator(bernfrac(2*n)/(-4*n))); \\ _Michel Marcus_, Sep 10 2013

%o (Magma) [1] cat [Denominator(Bernoulli(2*n)/(-4*n)):n in [1..35]]; // _G. C. Greubel_, Sep 19 2019

%o (Sage) [1]+[denominator(bernoulli(2*n)/(-4*n)) for n in (1..35)] # _G. C. Greubel_, Sep 19 2019

%o (GAP) Concatenation([1], List([1..35], n-> DenominatorRat(Bernoulli(2*n)/(-4*n)) )); # _G. C. Greubel_, Sep 19 2019

%Y Numerators are A001067.

%Y Cf. A000367/A002445, A002322, A079612.

%K nonn,easy,frac,nice

%O 0,2

%A _N. J. A. Sloane_, _Jeffrey Shallit_, _Simon Plouffe_

%E Thanks to _Michael Somos_ for helpful comments.