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.

2, 26861, 616897, 616909, 616933, 623641, 623653, 623669, 623681, 12315529, 12315581, 12315613, 12315617, 12362653, 12362657, 12362717, 12362741, 12362981, 12362989, 12365033, 12365057, 12365153, 12365173, 12365201, 12366589, 951821281

Offset: 1

Author

Gerhard Kirchner, Nov 20 2021

Keywords

• nonn

Comments

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.

Note the gaps between 2, 26861 and 616897, 623681 and 12315529, 12366589 and 951821281.

Examples

			primes 4*j+1: 5, 13, 17, ...
       4*j+3: 3, 7, 11, ...
d(x) = pi(x,4,3) - pi(x,4,1)
.
  n     x  pi(x,4,3) pi(x,4,1)   d(x)=1-n?
  - -----  --------- ---------  -----------
  1     2       0         0      0=0  true   a(1)  = 2
  2     3       1         0      1=-1 false  a(2) != 3
  2     5       1         1      2=-1 false  a(2) != 5
  ...........................
  2 26861    1472      1473     -1=-1 true   a(3)  = 26861
		

Links

• Eric Weisstein's World of Mathematics, Prime Quadratic Effect.

Crossrefs

Cf. A038691, A007350.

Programs

Maxima

block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 25,
  /* returns nmax terms */
while n