

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

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(2k1)).
a(n) = Sum_{k=1..n} A036263(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 ..]
 Reinhard Zumkeller, Aug 02 2012


CROSSREFS

Cf. A008347 (alternating sum of primes), A036263 (second difference of primes).
Sequence in context: A263411 A352950 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



