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.

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

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