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A326897
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Prime numbers p such that 0 < pi(p;10,(9,1)) = pi(p;10,(3,9)) where pi(x;q,(a,b)) is the number of primes p_n <= x such that p_n == a (mod q) and p_(n+1) == b (mod q).
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2
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29, 59, 199, 239, 269, 3559, 3659, 3739, 3929, 4013, 5003, 5023, 5273, 5303, 5419, 5449, 5563, 5639, 6299, 8219, 10079, 10639, 10723, 10799, 10859, 10889, 11119, 11213, 11299, 11353, 11399, 11489, 11549, 11903, 11939, 11969, 12049, 19739, 19889, 19949, 19979, 20089, 20149, 20233, 20393, 20543, 20593, 21493
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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