

A098058


Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).


15



2, 3, 5, 11, 17, 23, 29, 31, 41, 47, 53, 59, 61, 71, 73, 83, 101, 107, 113, 131, 137, 139, 149, 151, 157, 167, 173, 179, 181, 191, 197, 227, 233, 239, 241, 251, 257, 263, 269, 271, 281, 283, 293, 311, 317, 331, 337, 347, 353, 367, 373, 383, 409, 419, 421, 431
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OFFSET

1,1


COMMENTS

First differences are also not divisible by 4.  Zak Seidov, Jun 23 2015
Starting with 3, group the primes into runs of consecutive primes either all == 1 (mod 4) or all == 3 (mod 4). Only the last prime of each run appears in this sequence. Since the runs alternate == 1 (mod 4) and == 3 (mod 4), so do the members of this sequence.  Franklin T. AdamsWatters, Jun 23 2015
The sequence is infinite, by Dirichlet's theorem on primes in arithmetic progressions. The sequence contains arbitrarily long gaps, by Daniel Shiu's theorem on strings of congruent primes (see A057619 and A057622). Conjecture: The sequence contains arbitrarily long strings of consecutive primes (see A289118).  Jonathan Sondow, Jun 25 2017


REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, A4.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Jens Kruse Andersen, Consecutive Congruent Primes


EXAMPLE

Prime(2) = 3, prime(3) = 5. 4 does not divide 53 so prime(2)=3 is in the sequence.
Runs: (3), (5), (7,11), (17), (19, 23), (29), (31), (37,41), (43,47), (53), ... The sequence is 2 followed by the last member of each run. Differences within each run are always divisible by 4.


MATHEMATICA

Prime[Select[Range[100], Mod[Prime[ # + 1]  Prime[ # ], 4] !=0 &]] (* Ray Chandler, Oct 09 2006 *)


PROG

(PARI) f(n) = for(x=1, n, z=(prime(x+1)prime(x)); if(z%4, print1(prime(x)", ")))
(PARI) alist(n)=my(r=vector(n), p=2, np, k=0); while(k<n, np=nextprime(p+1); if((npp)%4!=0, r[k++]=p); p=np); r \\ Franklin T. AdamsWatters, Jun 23 2015
(PARI) list(lim)=my(v=List(), p=2); forprime(q=3, nextprime(lim\1+1), if((qp)%4, listput(v, p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jun 24 2015


CROSSREFS

Cf. A098059, A289118.
Sequence in context: A049596 A049571 A263090 * A040054 A093503 A040036
Adjacent sequences: A098055 A098056 A098057 * A098059 A098060 A098061


KEYWORD

easy,nonn


AUTHOR

Cino Hilliard, Sep 11 2004


EXTENSIONS

Edited by Ray Chandler, Oct 26 2006


STATUS

approved



