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%I #21 Feb 14 2026 13:23:19
%S 3,17,1259,270269,116396279,103515091679,313986271960080719,
%T 3987804966219837013431179,1164668352920931987761876342466557189,
%U 141150686467724967796377879843703940661695849,484356786477979939357984922968235281751492257310909570867909780039
%N Primes of the form (A002110(k) + A002110(k + 1)) / 2 - 1.
%C Are there infinitely many primes q = (A002110(k) + A002110(k + 1)) / 2 - 1 such that q is the lesser of twin primes (A001359) ? The first such primes q in this sequence are {3, 17, 270269, 103515091679}.
%e p = 17: (A002110(2) + A002110(3)) / 2 - 1 = (6 + 30) / 2 - 1 = 17, thus p = 17 is a term.
%t seq[max_] := With[{p = FoldList[Times, Prime[Range[max]]]}, Select[(Most[p] + Rest[p])/2 - 1, PrimeQ]]; seq[40] (* _Amiram Eldar_, Feb 09 2026 *)
%Y Cf. A000040, A001359, A002110.
%K nonn
%O 1,1
%A _Ctibor O. Zizka_, Feb 09 2026