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A098044 Odd primes p such that Pi_{3,1}(p) = Pi_{3,2}(p) - 1, where Pi_{m,n}(p) denotes the number of primes q <= p with q == n (mod m).

%I A098044 #37 May 20 2021 22:55:20
%S A098044 3,7,13,19,37,43,79,163,223,229,608981812891,608981812951,
%T A098044 608981812993,608981813507,608981813621,608981813819,608981813837,
%U A098044 608981813861,608981813929,608981813941,608981814019,608981814143,608981814247,608981814823
%N A098044 Odd primes p such that Pi_{3,1}(p) = Pi_{3,2}(p) - 1, where Pi_{m,n}(p) denotes the number of primes q <= p with q == n (mod m).
%C A098044 This is the break-even point among the odd primes of the form 3n+1 versus primes the form 3n+2.
%C A098044 "On the average Pi_{3,2}(x) - Pi_{3,1}(x) is asymptotically sqrt(x)/Log(x). However, Hudson (with the help of Schinzel) showed in 1985 that lim_{x --> inf} (Pi_{3,2}(x) - Pi_{3,1}(x))/ sqrt(x)/Log(x) does not exist (in particular, it is not equal to 1)." [Ribenboim, p. 275.]
%C A098044 From _M. F. Hasler_, May 10 2021: (Start)
%C A098044 Using the data a(1..85509) computed by D. Johnson, the major gaps are as follows:
%C A098044          between                      and               size of the gap
%C A098044       a(10) = 229                a(11) = 608981812891       609e9
%C A098044    a(11491) = 609340824721    a(11492) = 610704087667       1.3e9
%C A098044    a(21325) = 610968213803    a(21326) = 6148171711471      5.5e12
%C A098044    a(38653) = 6148988909519   a(38654) = 6149773241849      784e6
%C A098044    a(49417) = 6151116531611   a(49418) = 6151907045033      791e6
%C A098044    a(65479) = 6152794922413   a(65480) = 6153794890993      1.0e9
%C A098044    a(73171) = 6154352395729   a(73172) = 6155140151519      788e6
%C A098044    a(85509) = 6156051951809           ???                   ???.   (End)
%D A098044 P. Ribenboim, The New Book of Prime Number Records, Springer-Verlag, NY, 1995, page 274.
%H A098044 Donovan Johnson, <a href="/A098044/b098044.txt">Table of n, a(n) for n = 1..85509</a>
%F A098044 For n>1, a(n) = A000040(A096629(n-1)).
%e A098044 There are five odd primes <= 37 of the form 3n+1. They are 7, 13, 19, 31, 37. There are five odd primes <= 37 of the form 3n+2. They are 5, 11, 17, 23, 29. Therefore 37 is a "break-even" point among the odd primes.
%t A098044 p31 = p32 = 0; lst = {}; Do[p = Prime[n]; Switch[ Mod[p, 3], 1, p31++, 2, p32++ ]; If[ p31==p32, AppendTo[lst, p]], {n, 3, 10^7}]; lst (* _Robert G. Wilson v_, Sep 11 2004 *)
%o A098044 (PARI) N=100; c=1; forprime(p=3,, if(p%3>1,c++,c--)||print1(p",")||N--||break) \\ Takes only ~1 second up to 1e8, but to see the next terms, beyond 6e11, replace p=3 with p=608981812891. - _M. F. Hasler_, May 10 2021
%Y A098044 Cf. A007352.
%K A098044 nonn
%O A098044 1,1
%A A098044 _Wayne VanWeerthuizen_, Sep 10 2004
%E A098044 Edited and extended by _Robert G. Wilson v_, Sep 11 2004
%E A098044 Initial entry 3 added by _David Wasserman_, Nov 07 2007
%E A098044 Edited and terms a(11) onward added by _Max Alekseyev_, Feb 09 2011