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

Author

Zhi-Wei Sun, Feb 24 2014

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A000720, A238281.

Programs

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}]