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A121573 Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n). 1


%S 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 29,33,31,35,33,43,47,49,51,51,53,49,51,59,63,65,61,63,59,63,65,55,43,

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

%C 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 T. D. Noe, <a href="/A121573/b121573.txt">Table of n, a(n) for n=1..10000</a>

%H T. D. Noe, <a href="http://www.sspectra.com/math/A121573.gif">Plot of 10^6 terms</a>

%F a(n) = Sum_{k=1..n} (prime(2k+1) - prime(2k)) - Sum_{k=1..n} (prime(2k) - prime(2k-1)).

%F a(n) = Sum_{k=1..n} A036263(2k-1).

%F a(n) = prime(2n+1) - 2*A008347(2n) - 2.

%e 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 s=0; Table[s=s+Prime[2n-1]+Prime[2n+1]-2*Prime[2n], {n,115}]

%t With[{g=Transpose[Differences/@Partition[Prime[Range[400]],3,2]]}, Accumulate[g[[2]]]-Accumulate[g[[1]]]](* _Harvey P. Dale_, May 28 2013 *)

%o (Haskell)

%o a121573 n = a121573_list !! (n-1)

%o a121573_list = scanl1 (+) $ map a036263 [1, 3 ..]

%o -- _Reinhard Zumkeller_, Aug 02 2012

%Y Cf. A008347 (alternating sum of primes), A036263 (second difference of primes).

%K nice,sign,look

%O 1,2

%A _T. D. Noe_, Aug 08 2006

%E Typo in Formula fixed by _Reinhard Zumkeller_, Aug 02 2012

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Last modified March 2 09:56 EST 2021. Contains 341746 sequences. (Running on oeis4.)