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