cp's OEIS Frontend

A121573 Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n).

%I A121573 #18 Dec 21 2021 23:27:47
%S A121573 1,3,5,7,3,1,3,3,7,5,3,5,3,1,11,13,21,25,23,23,31,33,43,35,37,33,29,
%T A121573 29,33,31,35,33,43,47,49,51,51,53,49,51,59,63,65,61,63,59,63,65,55,43,
%U A121573 39,35,39,39,43,41,51,43,45,41,33,35,33,31,31,35,33,29,25,15,7,5,9,7,17,15,31,35,33,35,43,45,47,53,55,63,67,59,51,63,61
%N A121573 Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n).
%C A121573 This sequence was inspired by seeing two lines in the plot of A008347. It was expected that, on average, the gaps above prime(2n) would be larger than the gaps below prime(2n) and hence a(n) would be a mostly positive sequence. With some exceptions, this is true for the first 6330 terms. However, as the plot shows, over 500000 negative terms follow!
%H A121573 T. D. Noe, <a href="/A121573/b121573.txt">Table of n, a(n) for n = 1..10000</a>
%H A121573 T. D. Noe, <a href="http://www.sspectra.com/math/A121573.gif">Plot of 10^6 terms</a>
%F A121573 a(n) = Sum_{k=1..n} (prime(2k+1) - prime(2k)) - Sum_{k=1..n} (prime(2k) - prime(2k-1)).
%F A121573 a(n) = Sum_{k=1..n} A036263(2k-1).
%F A121573 a(n) = prime(2n+1) - 2*A008347(2n) - 2.
%e A121573 a(6)=1 because the prime gaps above and below the even-indexed primes (3,7,13,19,29,37) are 2,4,4,4,2,4 and 1,2,2,2,6,6, respectively. The sums of these gaps are 20 and 19, which differ by 1.
%t A121573 s=0; Table[s=s+Prime[2n-1]+Prime[2n+1]-2*Prime[2n], {n,115}]
%t A121573 With[{g=Transpose[Differences/@Partition[Prime[Range[400]],3,2]]}, Accumulate[g[[2]]]-Accumulate[g[[1]]]](* _Harvey P. Dale_, May 28 2013 *)
%o A121573 (Haskell)
%o A121573 a121573 n = a121573_list !! (n-1)
%o A121573 a121573_list = scanl1 (+) $ map a036263 [1, 3 ..]
%o A121573 -- _Reinhard Zumkeller_, Aug 02 2012
%Y A121573 Cf. A008347 (alternating sum of primes), A036263 (second difference of primes).
%K A121573 nice,sign,look
%O A121573 1,2
%A A121573 _T. D. Noe_, Aug 08 2006
%E A121573 Typo in Formula fixed by _Reinhard Zumkeller_, Aug 02 2012