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Search: id:A121573
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| A121573 |
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Prime-gap race; difference of the cumulative sums of gaps above and below prime(2n). |
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+0
1
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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!
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
T. D. Noe, Plot of 10^6 terms
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FORMULA
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Three: 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); a(n)=prime(2n+1)-2*A008347(2n)-2
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EXAMPLE
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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.
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MATHEMATICA
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s=0; Table[s=s+Prime[2n-1]+Prime[2n+1]-2*Prime[2n], {n, 115}]
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CROSSREFS
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Cf. A008347 (alternating sum of primes), A036263 (second difference of primes).
Sequence in context: A084763 A131214 A104260 this_sequence A130140 A051417 A090368
Adjacent sequences: A121570 A121571 A121572 this_sequence A121574 A121575 A121576
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KEYWORD
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nice,sign
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Aug 08 2006
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