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

3, 7, 13, 19, 43, 103, 109, 193, 229, 313, 349, 401, 463, 491, 509, 643, 743, 761, 823, 859, 883, 911, 997, 1093, 1237, 1279, 1303, 1429, 1459, 1483, 1489, 1499, 1571, 1609, 1637, 1831, 1873, 1999, 2003, 2069, 2083, 2221, 2239, 2243, 2251, 2269, 2273, 2399

Offset: 1

Author

Juri-Stepan Gerasimov, Aug 29 2011

Keywords

• nonn

Comments

Primes prime(k) such that A062234(k) = A062234(k-1). - Thomas Ordowski, Jan 03 2016

Primes prime(k) such that A001223(k) = 2*A001223(k-1). - Robert Israel, Jan 03 2016

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

Examples

			a(1)=3 (=(2*2+5)/3), a(2)=7 (=(2*5+11)/3), a(3)=13 (=(2*11+17)/3).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001223, A062234.

Programs

Maple

Primes:= select(isprime, [2,seq(i,i=3..10^4,2)]):
Gaps:= Primes[2..-1]-Primes[1..-2]:
Primes[select(t -> 2*Gaps[t-1] = Gaps[t],[$2..nops(Gaps)])]; # Robert Israel, Jan 03 2016

Mathematica

Table[(2 Prime[k - 1] + Prime[k + 1])/3, {k, 2, 360}] /. {Rational -> Nothing, n /; CompositeQ@ n -> Nothing} (* Michael De Vlieger, Jan 09 2016 *)

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

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

Extensions

Entries corrected by R. J. Mathar, Sep 30 2011