

A121573


Primegap race; difference of the cumulative sums of gaps above and below prime(2n).


1



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
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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



FORMULA

a(n) = Sum_{k=1..n} (prime(2k+1)  prime(2k))  Sum_{k=1..n} (prime(2k)  prime(2k1)).
a(n) = prime(2n+1)  2*A008347(2n)  2.


EXAMPLE

a(6)=1 because the prime gaps above and below the evenindexed 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[2n1]+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 !! (n1)
a121573_list = scanl1 (+) $ map a036263 [1, 3 ..]


CROSSREFS

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


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



