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A177961 a(1)=2. Otherwise the average of the smallest prime divisors of 2n-1 and 2n+1.

%I A177961 #22 Sep 08 2022 08:45:53
%S A177961 2,4,6,5,7,12,8,10,18,11,13,14,4,16,30,17,4,21,20,22,42,23,25,27,5,28,
%T A177961 29,4,31,60,32,4,36,35,37,72,38,5,43,41,43,44,4,46,48,5,4,51,50,52,
%U A177961 102,53,55,108,56,58,59,4,5,9,7,4,66,65,67,69,5,70,138,71,7,8,4,76,150,77,4,81
%N A177961 a(1)=2. Otherwise the average of the smallest prime divisors of 2n-1 and 2n+1.
%C A177961 As n tends to infinity, we have 1) lim inf (a(n)/n)=0; 2) if there exist infinitely many twin primes, then lim sup (a(n)/n)=2, otherwise, lim sup (a(n)/n)=1.
%H A177961 Robert Israel, <a href="/A177961/b177961.txt">Table of n, a(n) for n = 1..10000</a>
%F A177961 a(n) = (A090368(n)+A090368(n+1))/2. [_R. J. Mathar_, May 31 2010]
%p A177961 N:= 100: # to get a(1) to a(N)
%p A177961 S:= [1,seq(min(numtheory:-factorset(2*i-1)),i=2..N+1)]:
%p A177961 (S[2..-1]+S[1..-2])/2; # _Robert Israel_, Jul 31 2015
%t A177961 Table[If[n == 1, 2, Mean[{FactorInteger[2 n - 1][[1, 1]], FactorInteger[2 n + 1][[1, 1]]}]], {n, 78}] (* _Michael De Vlieger_, Aug 02 2015 *)
%o A177961 (PARI) a(n) = if (n==1, 2, (vecmin(factor(2*n-1)[,1]) + vecmin(factor(2*n+1)[,1]))/2); \\ _Michel Marcus_, Feb 07 2016
%o A177961 (Magma) [2] cat [1/2*(Min(PrimeFactors(2*n-1))+ Min(PrimeFactors(2*n+1))):n in [2..80]]; // _Vincenzo Librandi_, Feb 07 2016
%Y A177961 Cf. A177930, A177964, A177965, A177966.
%K A177961 nonn,easy
%O A177961 1,1
%A A177961 _Vladimir Shevelev_, May 16 2010, May 22 2010
%E A177961 More terms from _R. J. Mathar_, May 31 2010