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A194581 Primes prime(k) of the form (2*prime(k-1) + prime(k+1))/3.

%I A194581 #38 Mar 11 2016 06:22:02
%S A194581 3,7,13,19,43,103,109,193,229,313,349,401,463,491,509,643,743,761,823,
%T A194581 859,883,911,997,1093,1237,1279,1303,1429,1459,1483,1489,1499,1571,
%U A194581 1609,1637,1831,1873,1999,2003,2069,2083,2221,2239,2243,2251,2269,2273,2399
%N A194581 Primes prime(k) of the form (2*prime(k-1) + prime(k+1))/3.
%C A194581 Primes prime(k) such that A062234(k) = A062234(k-1). - _Thomas Ordowski_, Jan 03 2016
%C A194581 Primes prime(k) such that A001223(k) = 2*A001223(k-1). - _Robert Israel_, Jan 03 2016
%C A194581 Or, primes which are at 1/3 of the distance between the previous and next prime. See A267291 for primes which are at 2/3 between their neighbors. - _M. F. Hasler_, Jan 12 2016
%H A194581 Robert Israel, <a href="/A194581/b194581.txt">Table of n, a(n) for n = 1..10000</a>
%e A194581 a(1)=3 (=(2*2+5)/3), a(2)=7 (=(2*5+11)/3), a(3)=13 (=(2*11+17)/3).
%p A194581 Primes:= select(isprime, [2,seq(i,i=3..10^4,2)]):
%p A194581 Gaps:= Primes[2..-1]-Primes[1..-2]:
%p A194581 Primes[select(t -> 2*Gaps[t-1] = Gaps[t],[$2..nops(Gaps)])]; # _Robert Israel_, Jan 03 2016
%t A194581 Table[(2 Prime[k - 1] + Prime[k + 1])/3, {k, 2, 360}] /. {_Rational -> Nothing, n_ /; CompositeQ@ n -> Nothing} (* _Michael De Vlieger_, Jan 09 2016 *)
%o A194581 (PARI) for(k=2, 1000, q=2*prime(k-1)+prime(k+1); if(q%3==0 && isprime(q\3), print1(q\3, ", "))) \\ _Colin Barker_, Jun 27 2014
%o A194581 (PARI) A194581(n,show=0,o=2,g=0)={forprime(p=o+1,,g*2==(g=-o+o=p)||next; show&&print1(p-g",");n--||return(p-g))} \\ 2nd & 3rd optional args allow printing the whole list and using another starting value. - _M. F. Hasler_, Jan 12 2016
%Y A194581 Cf. A000040, A001223, A062234.
%K A194581 nonn
%O A194581 1,1
%A A194581 _Juri-Stepan Gerasimov_, Aug 29 2011
%E A194581 Entries corrected by _R. J. Mathar_, Sep 30 2011