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Search: id:A092243
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| A092243 |
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Score at stage n for "tug of war" between prime gap increases vs. prime gap decreases: start with score = 0 at n = 1 and at stage n = k > 1, increase (resp. decrease) the score by 1 if the k-th prime gap is greater (resp. less) than the previous prime gap. |
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+0
1
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| 0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 3, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 1, 2, 3, 2, 3, 2, 2, 2, 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 3, 2, 3, 4, 3, 4, 5, 4, 5, 4, 5, 4, 5, 6, 5, 4, 5, 6, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 5, 4, 5
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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a(n) is nonnegative for n = 1,...,41,252. At n = 41253, a(n) = -1. At most larger values of n, up to n = 250000 (as far as I've checked), a(n) is overwhelmingly negative.
Questions. Is s > 0 for some n > 250000? Is s bounded from below? Is s bounded from above? Is s > 0 for infinitely many values of n? Is s < 0 for infinitely many values of n?
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LINKS
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Pe, J. L. Prime Gap Tug of War
C. Rivera, Prime Puzzles: the prime gap tug of war answers some but not all of the questions in the comments above.
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EXAMPLE
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At stage n = 1, the score a(1) = 0. The first prime gap is 3-2 = 1.
At stage n = 2, the second prime gap is 5-3 = 2 > 1, the previous prime gap. Hence a(2) = a(1) + 1 = 0 + 1 = 1.
At stage n = 3, the third prime gap is 7-5 = 2, which equals the previous prime gap. The score doesn't change; hence a(3) = 1.
At stage n = 4, the fourth prime gap is 11-7 = 4 > 2, the third prime gap. Hence a(4) = a(3) + 1 = 1+1 = 2.
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MATHEMATICA
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d = 1; c = 3; s = 0; r = {0}; For[i = 2, i <= 200, i++, e = Prime[i + 1]; newd = e - c; c = e; If[newd > d, s = s + 1, If[newd < d, s = s - 1]]; d = newd; r = Append[r, s]]; r
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CROSSREFS
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Sequence in context: A060272 A129985 A085243 this_sequence A054716 A060145 A035391
Adjacent sequences: A092240 A092241 A092242 this_sequence A092244 A092245 A092246
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KEYWORD
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nonn
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AUTHOR
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Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Feb 19 2004
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