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A274828
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Integer part of the alternating n-th partial sum of the reciprocals of the successive prime gaps.
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4
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1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
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OFFSET
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1,41
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COMMENTS
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The graph of the first 15*10^3 terms looks like a realization of a random walk. It has an estimated fractal dimension of about 1.48 (with box counting method) which is closed to that of the random walk (3/2).
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LINKS
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FORMULA
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a(n) = floor(Sum_{i=1..n} ((-1)^(i - 1))/(prime(i+1)-prime(i))).
a(n) = floor(Sum_{i=1..n} ((-1)^(i - 1))/A001223(i)).
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EXAMPLE
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The prime gaps (A001223) are 1, 2, 2, 4, 2, 4, 2, .... For n=7, the 7th partial sum is 1/1 - 1/2 + 1/2 - 1/4 + 1/2 - 1/4 + 1/2 = 3/2 so a(7) is the integer part of 3/2, which is 1. - Michael B. Porter, Jul 11 2016
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MATHEMATICA
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Table[Floor@Sum[((-1)^(j - 1))/(Prime[j + 1] - Prime[j]), {j, 1, n}], {n, 1, 100}];
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PROG
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(PARI) a(n) = floor(sum(i=1, n, ((-1)^(i-1))/(prime(i+1)-prime(i)))) \\ Felix Fröhlich, Jul 07 2016
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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