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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
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
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
Sequence in context: A184310 A301347 A241811 * A155187 A338613 A109132
KEYWORD
nonn
AUTHOR
Gerhard Kirchner, Nov 20 2021
STATUS
approved