Largest prime in the partials sums of primes in A134125 which have integer averages.

Terms

a(0) =3a(1) =5a(2) =11a(3) =19a(4) =31a(5) =233a(6) =739a(7) =2207a(8) =4871a(9) =47933a(10) =76103a(11) =82723a(12) =128663a(13) =391273a(14) =521041a(15) =769423a(16) =2036833a(17) =3724997a(18) =14722933a(19) =31957817a(20) =87574217a(21) =167518933a(22) =478372393a(23) =656640899a(24) =749613233a(25) =861934273a(26) =9083114473a(27) =29862785453a(28) =95892456511a(29) =160534630967