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 A092243 Score at stage n in "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. 6
 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; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS a(n) is nonnegative for n = 1,...,41252. 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? LINKS N. J. A. Sloane, Table of n, a(n) for n = 1..20000 Pe, J. L., Prime Gap Tug of War, 2002 Pe, J. L., Prime Gap Tug of War, 2002 [Cached copy, pdf file only, with permission.] Shows extended graphs. C. Rivera, Puzzle 271: Prime gap tug of war. N. J. A. Sloane, Table of n, a(n) for n = 1..100000 N. J. A. Sloane, Table of n, a(n) for n = 1..965562 FORMULA Cumulative sums of A079054 (negated). EXAMPLE 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. MAPLE # From N. J. A. Sloane, Mar 13 2016 (a is A079054, ss is the present sequence): a:=[]; ss:=[0]; s:=0; M:=120; for n from 2 to M-1 do q:=ithprime(n); p:=prevprime(q); r:=nextprime(q); if q-p < r-q then a:=[op(a), -1]; s:=s+1; elif q-p=r-q then a:=[op(a), 0]; else a:=[op(a), 1]; s:=s-1; fi; ss:=[op(ss), s]; od: a; ss; MATHEMATICA 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 CROSSREFS Cf. A079054. For indices where there is a strict sign change see A269737. For positions of records see A269738, A269739. Positions of zeros: A175102. Sequence in context: A129985 A085243 A265745 * A257564 A194509 A054716 Adjacent sequences:  A092240 A092241 A092242 * A092244 A092245 A092246 KEYWORD sign AUTHOR Joseph L. Pe, Feb 19 2004 STATUS approved

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Last modified November 30 06:11 EST 2020. Contains 338781 sequences. (Running on oeis4.)