OFFSET
1,1
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.
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
EXAMPLE
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.
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}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 22 2014
STATUS
approved