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

%I A098058 #21 Jun 26 2017 04:10:23
%S A098058 2,3,5,11,17,23,29,31,41,47,53,59,61,71,73,83,101,107,113,131,137,139,
%T A098058 149,151,157,167,173,179,181,191,197,227,233,239,241,251,257,263,269,
%U A098058 271,281,283,293,311,317,331,337,347,353,367,373,383,409,419,421,431
%N A098058 Prime(n) such that 4 does not divide the difference between prime(n) and prime(n+1).
%C A098058 First differences are also not divisible by 4. - _Zak Seidov_, Jun 23 2015
%C A098058 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
%C A098058 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
%D A098058 R. K. Guy, Unsolved Problems in Number Theory, A4.
%H A098058 Charles R Greathouse IV, <a href="/A098058/b098058.txt">Table of n, a(n) for n = 1..10000</a>
%H A098058 Jens Kruse Andersen, <a href="http://primerecords.dk/congruent-primes.htm">Consecutive Congruent Primes</a>
%e A098058 Prime(2) = 3, prime(3) = 5. 4 does not divide 5-3 so prime(2)=3 is in the sequence.
%e A098058 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.
%t A098058 Prime[Select[Range[100], Mod[Prime[ # + 1] - Prime[ # ], 4] !=0 &]] (* _Ray Chandler_, Oct 09 2006 *)
%o A098058 (PARI) f(n) = for(x=1,n,z=(prime(x+1)-prime(x));if(z%4,print1(prime(x)",")))
%o A098058 (PARI) alist(n)=my(r=vector(n),p=2,np,k=0);while(k<n,np=nextprime(p+1);if((np-p)%4!=0,r[k++]=p);p=np);r \\ _Franklin T. Adams-Watters_, Jun 23 2015
%o A098058 (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
%Y A098058 Cf. A098059, A289118.
%K A098058 easy,nonn
%O A098058 1,1
%A A098058 _Cino Hilliard_, Sep 11 2004
%E A098058 Edited by _Ray Chandler_, Oct 26 2006