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

Author

T. D. Noe, Aug 08 2006

Keywords

• nice
• sign
• look

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!

Examples

			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.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• T. D. Noe, Plot of 10^6 terms

Crossrefs

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

Programs

Haskell

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

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 *)

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.

Extensions

Typo in Formula fixed by Reinhard Zumkeller, Aug 02 2012