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A349519 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = 1-n where pi(x,4,k) is the number of primes 4*j + k <= x.

%I A349519 #19 Feb 16 2025 08:34:02
%S A349519 2,26861,616897,616909,616933,623641,623653,623669,623681,12315529,
%T A349519 12315581,12315613,12315617,12362653,12362657,12362717,12362741,
%U A349519 12362981,12362989,12365033,12365057,12365153,12365173,12365201,12366589,951821281
%N A349519 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = 1-n where pi(x,4,k) is the number of primes 4*j + k <= x.
%C A349519 The difference d(x) = pi(x,4,3) - pi(x,4,1) changes sign infinitely often, see link "Prime Quadratic Effect". But this does not say anything about the amplitudes of these oscillations. For diagrams, see A349518, "Oscillations of d(x)". If d(x) has no lower limit, the current sequence is infinite. Regarding the upper limit, see A349518.
%C A349519 Note the gaps between 2, 26861 and 616897, 623681 and 12315529, 12366589 and 951821281.
%H A349519 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeQuadraticEffect.html">Prime Quadratic Effect</a>.
%e A349519 primes 4*j+1: 5, 13, 17, ...
%e A349519        4*j+3: 3, 7, 11, ...
%e A349519 d(x) = pi(x,4,3) - pi(x,4,1)
%e A349519 .
%e A349519   n     x  pi(x,4,3) pi(x,4,1)   d(x)=1-n?
%e A349519   - -----  --------- ---------  -----------
%e A349519   1     2       0         0      0=0  true   a(1)  = 2
%e A349519   2     3       1         0      1=-1 false  a(2) != 3
%e A349519   2     5       1         1      2=-1 false  a(2) != 5
%e A349519   ...........................
%e A349519   2 26861    1472      1473     -1=-1 true   a(3)  = 26861
%o A349519 (Maxima) block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 25,
%o A349519   /* returns nmax terms */
%o A349519 while n<nmax do(
%o A349519   p: next_prime(p), su:su+mod(p,4)-2,
%o A349519   if su<sum then(n:n+1, sum:su, w: append(w,[p]) ) ) ,
%o A349519 w);
%Y A349519 Cf. A038691, A007350.
%K A349519 nonn
%O A349519 1,1
%A A349519 _Gerhard Kirchner_, Nov 20 2021