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

1,1

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.

Links

Table of n, a(n) for n=1..26.

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

Example

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

Prog

(Maxima) block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 25,
  /* returns nmax terms */
while n<nmax do(
  p: next_prime(p), su:su+mod(p, 4)-2,
  if su<sum then(n:n+1, sum:su, w: append(w, [p]) ) ) ,
w);

Crossrefs

Cf. A038691, A007350.

Sequence in context: A237521 A131558 A133857 * A232869 A153912 A247790

Adjacent sequences: A349516 A349517 A349518 * A349520 A349521 A349522

Keyword

nonn

Author

Gerhard Kirchner, Nov 20 2021

Status

approved