A210210 Least number k among 1, ..., n such that pi(p+2*n) - pi(p+n) = pi(p+n) - pi(p) > 0 with p = prime(k), or 0 if such a number k does not exist, where pi(.) is given by A000720.

0, 1, 2, 4, 2, 2, 4, 1, 2, 3, 3, 6, 2, 9, 4, 6, 7, 4, 4, 6, 3, 3, 3, 5, 2, 6, 4, 4, 4, 8, 3, 4, 6, 3, 3, 8, 8, 6, 6, 7, 7, 10, 7, 6, 14, 5, 5, 8, 5, 4, 6, 3, 3, 13, 2, 14, 12, 12, 12, 18, 18, 18, 11, 11, 11, 17, 10, 11, 11, 16, 16, 9, 9, 16, 15, 16, 8, 14, 14, 14

Offset

1,3

Comments

Conjecture: a(n) > 0 for all n > 1.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Example

a(4) = 4 since prime(4) = 7 with pi(7+2*4) - pi(7+4) = pi(7+4) - pi(7) = 1 > 0, but pi(2+2*4) - pi(2+4) = 1 < pi(2+4) - pi(2) = 2, pi(3+2*4) - pi(3+4) = 1 < pi(3+4) - pi(3) = 2, and pi(5+2*4) - pi(5+4) = 2 > pi(5+4) - pi(5) = 1.

Mathematica

p[k_, n_]:=Prime[k]+n>=Prime[k+1]&&k+PrimePi[Prime[k]+2n]==2*PrimePi[Prime[k]+n]
Do[Do[If[p[k, n], Print[n, " ", k]; Goto[aa]], {k, 1, n}];
Print[n, " ", 0]; Label[aa]; Continue, {n, 1, 80}]

Crossrefs

Cf. A000040, A000720, A238281.

Sequence in context: A183193 A331470 A021809 * A117007 A340129 A213433

Adjacent sequences: A210207 A210208 A210209 * A210211 A210212 A210213

Keyword

nonn

Author

Zhi-Wei Sun, Feb 24 2014

Status

approved