login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A121573 Prime-gap 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 (list; graph; refs; listen; history; text; internal format)
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: A104260 A263792 A263411 * A196407 A156030 A255562

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 21 12:01 EDT 2017. Contains 290864 sequences.