OFFSET
1,1
COMMENTS
The counting function pi(x;q,(a,b)) is defined by Lemke Oliver and Soundararajan to be the number of primes p_n <= x such that p_n == a (mod q) and p_(n+1) == b (mod q).
Note that it is known that assertions like pi(x;3,1) < pi(x;3,2) or pi(x;4,1) < pi(x;4,3) ("Chebychev's biases") are false infinitely often.
Lemke Oliver and Soundararajan conjectured that pi(x;3,(1,1)) < pi(x,3,(1,2)) and pi(x;4,(1,1)) < pi(x;4,(1,3)) are robust biases, i.e., they always hold.
Here we conjecture that pi(x;10,(9,1)) > pi(x;10,(3,9)) is a semi- robust bias, i.e., it holds from a certain value x_0 on. Here x_0 is conjectured to be 48539. This conjecture seems to be confirmed by the observation that pi(x;10,(9,1)) - pi(x;10,(3,9)) ~ (Li(x)/16) * (1/2) * log(2*Pi/5*log(x)) / log(x).
LINKS
A.H.M. Smeets, Table of n, a(n) for n = 1..80
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, arXiv:1603.03720 [math.NT], 2016.
R. J. Lemke Oliver and K. Soundararajan, Unexpected biases in the distribution of consecutive primes, Proceedings of the National Academy of Sciences of the United States of America, Vol. 113, No. 31 (2016), E4446-E4454.
EXAMPLE
First prime p such that (p,p') == (9,1) (mod 10) is for p = 29.
First prime q such that (q,q') == (3,9) (mod 10) is for q = 23.
So pi(x;10,(9,1)) = pi(x;10,(3,9)) > 0 first occurs for x = 29, so a(1) = 29.
PROG
(PARI) lista(nn) = {my(vp = primes(nn)); my(nba = 0, nbb = 0); for (n=1, nn-1, my(ok = 0); my(mp = vp[n] % 10); my(mq = vp[n+1] % 10); if ([mp, mq] == [9, 1], nba++; ok=1); if ([mp, mq] == [3, 9], nbb++; ok=1); if (ok && nba && (nba == nbb), print1(vp[n], ", ")); ); } \\ Michel Marcus, Sep 14 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
A.H.M. Smeets, Sep 13 2019
STATUS
approved