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

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

Offset

1,2

Comments

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!

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

T. D. Noe, Plot of 10^6 terms

Formula

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

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

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

Example

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.

Mathematica

s=0; Table[s=s+Prime[2n-1]+Prime[2n+1]-2*Prime[2n], {n, 115}]
With[{g=Transpose[Differences/@Partition[Prime[Range[400]], 3, 2]]}, Accumulate[g[[2]]]-Accumulate[g[[1]]]](* Harvey P. Dale, May 28 2013 *)

Prog

(Haskell)
a121573 n = a121573_list !! (n-1)
a121573_list = scanl1 (+) $ map a036263 [1, 3 ..]
-- Reinhard Zumkeller, Aug 02 2012

Crossrefs

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

Sequence in context: A367291 A367289 A351463 * A196407 A156030 A338974

Adjacent sequences: A121570 A121571 A121572 * A121574 A121575 A121576

Keyword

nice,sign,look

Author

T. D. Noe, Aug 08 2006

Extensions

Typo in Formula fixed by Reinhard Zumkeller, Aug 02 2012

Status

approved