This site is supported by donations to The OEIS Foundation.

 Annual Appeal: Please make a donation to keep the OEIS running. In 2017 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A001897 Denominators of cosecant numbers -2*(2^(2*n-1)-1)*Bernoulli(2*n). (Formerly M2983 N1205) 17
 1, 3, 15, 21, 15, 33, 1365, 3, 255, 399, 165, 69, 1365, 3, 435, 7161, 255, 3, 959595, 3, 6765, 903, 345, 141, 23205, 33, 795, 399, 435, 177, 28393365, 3, 255, 32361, 15, 2343, 70050435, 3, 15, 1659, 115005, 249, 1702155, 3, 30705, 136059, 705, 3, 2250885, 3, 16665, 2163 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS ‎Same as half the denominators of the even-indexed Bernoulli numbers B_{2*n} for n>0, by the von Staudt-Clausen theorem and Fermat's little theorem.‎ - Bernd Kellner and Jonathan Sondow, Jan 02 2017 REFERENCES H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 187. S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. N. E. Nörlund, Vorlesungen über Differenzenrechnung. Springer-Verlag, Berlin, 1924, p. 458. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Hector Blandin and Rafael Diaz, Compositional Bernoulli numbers, arXiv:0708.0809 [math.CO], 2007-2008, Page 7, 3rd table, (B^sin)_1,n is identical to |A001896| / A001897. S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes] D. H. Lehmer, Lacunary recurrence formulas for the numbers of Bernoulli and Euler, Annals Math., 36 (1935), 637-649. N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer 1924, p. 27. N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463] FORMULA a(0)=1, a(n)=(1/2)*A002445(n) for n>=1. - Joerg Arndt, May 07 2012 a(n) = denominator((2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012 a(0)=1, a(n) = (1/2)*A027642(2*n) = (3/2)*A277087(n) for n>=1. - Jonathan Sondow, Dec 14 2016 From Peter Luschny, Sep 06 2017: (Start) a(n) = denominator(r(n)) where r(n) = Sum_{0..n} (-1)^(n-k)*A241171(n, k)/(2*k+1). a(n) = denominator(bernoulli(2*n, 1/2))/4^n = A033469(n)/4^n. (End) EXAMPLE Cosecant numbers {-2*(2^(2*n-1)-1)*Bernoulli(2*n)} are 1, -1/3, 7/15, -31/21, 127/15, -2555/33, 1414477/1365, -57337/3, 118518239/255, -5749691557/399, 91546277357/165, -1792042792463/69, 1982765468311237/1365, -286994504449393/3, 3187598676787461083/435, ... = A001896/A001897. MAPLE b := n -> bernoulli(n)*2^add(i, i=convert(n, base, 2)); a := n -> denom(b(2*n)); # Peter Luschny, May 02 2009 # Alternative : Clausen := proc(n) local i, S; map(i->i+1, numtheory[divisors](n)); S := select(isprime, %); if S <> {} then mul(i, i=S) else NULL fi end: A001897_list := n -> [1, seq(Clausen(2*i)/2, i=1..n-1)]; A001897_list(52); # Peter Luschny, Oct 03 2011 MATHEMATICA a[n_] := Denominator[-2*(2^(2*n-1)-1)*BernoulliB[2*n]]; Table[a[n], {n, 0, 51}] (* Jean-François Alcover, Sep 11 2013 *) PROG (Sage) def A001897(n):     if n == 0: return 1     M = map(lambda i: i+1, divisors(2*n))     return mul(filter(lambda s: is_prime(s), M))/2 print [A001897(n) for n in range(52)] # Peter Luschny, Feb 20 2016 CROSSREFS Cf. A001896, A027642, A033469, A132092-A132106, A160014, A241171, A277087. Sequence in context: A097571 A048087 A316751 * A074214 A036897 A129966 Adjacent sequences:  A001894 A001895 A001896 * A001898 A001899 A001900 KEYWORD nonn,frac AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 10 02:30 EST 2018. Contains 318036 sequences. (Running on oeis4.)