A349518 a(n)=x is the least prime with pi(x,4,3) - pi(x,4,1) = n-1 where pi(x,4,k) is the number of primes 4*j + k <= x.

2, 3, 11, 71, 83, 223, 227, 503, 751, 1531, 1543, 1571, 1579, 1583, 4127, 5147, 5171, 5179, 5651, 6211, 11083, 11087, 11471, 11483, 11519, 11527, 11579, 11587, 17239, 17551, 17903, 17971, 35963, 36011, 39703, 39727

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 link "Oscillations of d(x)". If d(x) has no upper limit, the current sequence is infinite. Regarding the lower limit, see A349519.

Links

Table of n, a(n) for n=1..36.

Gerhard Kirchner, Oscillations of d(x)

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)=n-1?
  - --  --------- ---------  ---------
  1  2       0         0     0=0  true   a(1)  = 2
  2  3       1         0     1=1  true   a(2)  = 3
  3  5       1         1     0=2 false   a(3) != 5
  ...........................
  3 11       3         1     2=2  true   a(3)  = 11

Prog

(Maxima) block(w:[2],  su:0, sum:0, n:1, p:2, nmax: 40,
  /* 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: A184310 A301347 A241811 * A155187 A338613 A109132

Adjacent sequences: A349515 A349516 A349517 * A349519 A349520 A349521

Keyword

nonn

Author

Gerhard Kirchner, Nov 20 2021

Status

approved