

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 kth 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
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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 32 = 1.
At stage n = 2, the second prime gap is 53 = 2 > 1, the previous prime gap. Hence a(2) = a(1) + 1 = 0 + 1 = 1.
At stage n = 3, the third prime gap is 75 = 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 117 = 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 M1 do
q:=ithprime(n); p:=prevprime(q); r:=nextprime(q);
if qp < rq then a:=[op(a), 1]; s:=s+1;
elif qp=rq then a:=[op(a), 0]; else a:=[op(a), 1]; s:=s1; 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



