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A006863
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Denominator of B_{2n}/(-4n), where B_m are the Bernoulli numbers.
(Formerly M5150)
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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (p.j.cameron(AT)qmul.ac.uk), Mar 25 2002
(-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
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).
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REFERENCES
| Bruce Berndt, Ramanujan's Notebooks Part II, Springer-Verlag; see Integrals and Asymptotic Expansions, p. 220.
R. D. Carmichael, Note on a new number theory function, Bull. Amer. Math. Soc. 16 (1909-10), 232-238.
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.)
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..10000
G. Everest, Y. Puri and T. Ward, Integer sequences counting periodic points
Michael Lugo, A little number theory problem [From Tanya Khovanova, Feb 21 2009]
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to Bernoulli numbers.
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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 (p.j.cameron(AT)qmul.ac.uk), Mar 25 2002
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MATHEMATICA
| a[n_] := Denominator[BernoulliB[2n]/(-4n)]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Mar 20 2011 *)
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CROSSREFS
| Numerators are A001067. Cf. A000367/A002445, A002322, A079612.
Sequence in context: A003264 A003272 A003245 * A052663 A192491 A167548
Adjacent sequences: A006860 A006861 A006862 * A006864 A006865 A006866
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KEYWORD
| nonn,easy,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jeffrey Shallit, Simon Plouffe (simon.plouffe(AT)gmail.com)
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EXTENSIONS
| Thanks to Michael Somos for helpful comments.
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