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
Keywords
Examples
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.
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
Crossrefs
Programs
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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;
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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
Formula
Cumulative sums of A079054 (negated).
Comments