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A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).

%I A051717 #43 Jul 02 2025 16:01:58
%S A051717 1,2,3,6,30,30,42,42,30,30,66,66,2730,2730,6,6,510,510,798,798,330,
%T A051717 330,138,138,2730,2730,6,6,870,870,14322,14322,510,510,6,6,1919190,
%U A051717 1919190,6,6,13530,13530,1806,1806,690,690,282,282,46410,46410,66,66,1590,1590
%N A051717 1, followed by denominators of first differences of Bernoulli numbers (B(i)-B(i-1)).
%C A051717 Equivalently, denominators of Bernoulli twin numbers C(n) (cf. A051716).
%C A051717 The Bernoulli twin numbers C(n) are defined by C(0) = 1, then C(2n) = B(2n) + B(2n-1), C(2n+1) = -B(2n+1) - B(2n), where B() are the Bernoulli numbers A027641/A027642. The definition is due to _Paul Curtz_.
%C A051717 Denominators of column 1 of table described in A051714/A051715.
%H A051717 G. C. Greubel, <a href="/A051717/b051717.txt">Table of n, a(n) for n = 0..5000</a>
%H A051717 M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.
%e A051717 Bernoulli numbers: 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ...
%e A051717 First differences: -3/2, 2/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, ...
%e A051717 Numerators: -3, 2, -1, -1, 1, 1, -1, -1, 1, 5, -5, -691, 691, 7, ...
%e A051717 Denominators: 2, 3, 6, 30, 30, 42, 42, 30, 30, 66, 66, 2730, ...
%e A051717 Sequence of C(n)'s begins: 1, -1/2, -1/3, -1/6, -1/30, 1/30, 1/42, -1/42, -1/30, 1/30, 5/66, -5/66, -691/2730, 691/2730, 7/6, -7/6, ...
%p A051717 C:=proc(n) if n=0 then RETURN(1); fi; if n mod 2 = 0 then RETURN(bernoulli(n)+bernoulli(n-1)); else RETURN(-bernoulli(n)-bernoulli(n-1)); fi; end;
%t A051717 c[0]= 1; c[n_?EvenQ]:= BernoulliB[n] + BernoulliB[n-1]; c[n_?OddQ]:= -BernoulliB[n] - BernoulliB[n-1]; Table[Denominator[c[n]], {n,0,53}] (* _Jean-François Alcover_, Dec 19 2011 *)
%t A051717 Join[{1},Denominator[Total/@Partition[BernoulliB[Range[0,60]],2,1]]] (* _Harvey P. Dale_, Mar 09 2013 *)
%t A051717 Join[{1},Denominator[Differences[BernoulliB[Range[0,60]]]]] (* _Harvey P. Dale_, Jun 28 2021 *)
%o A051717 (PARI) a(n)=if(n<3,n+1,denominator(bernfrac(n)-bernfrac(n-1))) \\ _Charles R Greathouse IV_, May 18 2015
%o A051717 (Magma)
%o A051717 f:= func< n | Bernoulli(n) + Bernoulli(n-1) >;
%o A051717 function A051717(n)
%o A051717   if n eq 0 then return 1;
%o A051717   elif (n mod 2) eq 0 then return Denominator(f(n));
%o A051717   else return Denominator(-f(n));
%o A051717   end if;
%o A051717 end function;
%o A051717 [A051717(n): n in [0..50]]; // _G. C. Greubel_, Apr 22 2023
%o A051717 (SageMath)
%o A051717 def f(n): return bernoulli(n)+bernoulli(n-1)
%o A051717 def A051717(n):
%o A051717     if (n==0): return 1
%o A051717     elif (n%2==0): return denominator(f(n))
%o A051717     else: return denominator(-f(n))
%o A051717 [A051717(n) for n in range(51)] # _G. C. Greubel_, Apr 22 2023
%Y A051717 Cf. A027641, A027642, A051714, A051715, A051716, A129825, A129826.
%Y A051717 Cf. A129724.
%Y A051717 For numerators see A172083.
%K A051717 nonn,easy,nice,frac
%O A051717 0,2
%A A051717 _N. J. A. Sloane_
%E A051717 More terms from _James Sellers_, Dec 08 1999
%E A051717 Edited by _N. J. A. Sloane_, May 25 2008
%E A051717 Entry revised by _N. J. A. Sloane_, Apr 22 2021