A006863 Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.

1, 24, 240, 504, 480, 264, 65520, 24, 16320, 28728, 13200, 552, 131040, 24, 6960, 171864, 32640, 24, 138181680, 24, 1082400, 151704, 5520, 1128, 4455360, 264, 12720, 86184, 13920, 1416, 6814407600, 24

Offset: 0

Author

N. J. A. Sloane, Jeffrey Shallit, Simon Plouffe

Keywords

• nonn
• easy
• frac
• nice

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.
• Index entries for sequences related to Bernoulli numbers.

Crossrefs

Numerators are A001067.

Cf. A000367/A002445, A002322, A079612.

Programs

GAP

Concatenation([1], List([1..35], n-> DenominatorRat(Bernoulli(2*n)/(-4*n)) )); # G. C. Greubel, Sep 19 2019

Magma

[1] cat [Denominator(Bernoulli(2*n)/(-4*n)):n in [1..35]]; // G. C. Greubel, Sep 19 2019

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 *)

PARI

a(n) = if (n == 0, 1, denominator(bernfrac(2*n)/(-4*n))); \\ Michel Marcus, Sep 10 2013

Sage

[1]+[denominator(bernoulli(2*n)/(-4*n)) for n in (1..35)] # G. C. Greubel, Sep 19 2019

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

Extensions

Thanks to Michael Somos for helpful comments.