A238289 Least positive integer k such that prime(k*n), prime((k+1)*n) and prime((k+2)*n) form an arithmetic progression, or 0 if such a number k does not exist.

2, 2, 17, 4, 1, 1, 59, 3, 56, 1, 39, 10, 9, 130, 2, 18, 11, 7, 80, 67, 2, 19, 27, 17, 92, 73, 180, 65, 5, 110, 282, 4, 6, 8, 16, 2, 23, 198, 20, 3, 99, 83, 217, 13, 110, 28, 16, 6, 5, 3, 144, 31, 9, 187, 176, 145, 75, 11, 43, 424, 4, 54, 272, 8, 26, 131, 123, 107, 8, 4, 52, 9, 127, 84, 264, 33, 145, 663, 16, 285

Offset: 1

Author

Zhi-Wei Sun, Feb 22 2014

Keywords

• nonn

Comments

Conjecture: (i) We always have 0 < a(n) < 3*prime(n) + 9.

(ii) For any integer n > 3, there is a positive integer k < 5*n^3 such that prime(k*n), prime((k+1)*n), prime((k+2)*n), and prime((k+3)*n) form a 4-term arithmetic progression.

(iii) In general, for each m = 3, 4, ..., if n is sufficiently large then there is a positive integer k = O(n^(m-1)) such that prime((k+j)*n) (j = 0, ..., m-1) form an arithmetic progression.

The conjecture is a refinement of the Green-Tao theorem.

Examples

			a(2) = 2 since prime(2*2) = 7, prime(3*2) = 13 and prime(4*2) = 19 form a 3-term arithmetic progression, but prime(1*2) = 3,  prime(2*2) = 7 and prime(3*2) = 13 do not form a 3-term arithmetic progression.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..8000
• B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Math. 167(2008), 481-547.
• Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Crossrefs

Cf. A000040, A238277, A238278, A238281.

Programs

Mathematica

d[k_,n_]:=Prime[(k+1)*n]-Prime[k*n]
Do[Do[If[d[k,n]==d[k+1,n],Print[n," ",k];Goto[aa]],{k,1,3*Prime[n]+8}];
Print[n," ",0];Label[aa];Continue,{n,1,80}]

PARI

okpr(p, q, r) = (q - p) == (r - q);
a(n) = {k = 1; while(! okpr(prime(k*n), prime((k+1)*n), prime((k+2)*n)), k++); k;} \\ Michel Marcus, Aug 28 2014