A098058 Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).

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

Offset: 1

Author

Cino Hilliard, Sep 11 2004

Keywords

• easy
• nonn

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. Adams-Watters, 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

Examples

			Prime(2) = 3, prime(3) = 5. 4 does not divide 5-3 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.
		

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

Crossrefs

Cf. A098059, A289118.

Programs

Mathematica

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

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(kFranklin T. Adams-Watters, Jun 23 2015

PARI

list(lim)=my(v=List(),p=2); forprime(q=3,nextprime(lim\1+1), if((q-p)%4, listput(v,p)); p=q); Vec(v) \\ Charles R Greathouse IV, Jun 24 2015

Extensions

Edited by Ray Chandler, Oct 26 2006