cp's OEIS Frontend

A098016 Indices x such that (1/4)(prime(x+1) + prime(x)) is prime.

%I A098016 #16 Sep 11 2022 14:37:29
%S A098016 2,3,9,11,23,32,54,58,67,76,86,103,164,188,200,202,208,210,243,311,
%T A098016 351,354,374,414,420,427,441,468,515,539,559,588,621,639,650,652,662,
%U A098016 670,693,708,748,752,769,811,816,823,842,883,889,939,943,963,970,1006,1009
%N A098016 Indices x such that (1/4)(prime(x+1) + prime(x)) is prime.
%C A098016 Conjecture: (1/2)(prime(x+1) + prime(x)) is not prime for all x.
%C A098016 This is obvious: (prime(x+1)+prime(x))/2 is strictly between prime(x) and prime(x+1), so if it were prime, prime(x+1) wouldn't be the next prime after prime(x). - _Robert Israel_, Feb 04 2019
%H A098016 Vincenzo Librandi, <a href="/A098016/b098016.txt">Table of n, a(n) for n = 1..1000</a>
%F A098016 a(n) = A000720(2*A118134(n)-1). - _Robert Israel_, Feb 04 2019
%e A098016 Prime(2+1) + prime(2) = 5+3 = 8. 1/4(8) = 2. 2 is the first entry.
%p A098016 filter:= proc(t) local v; v:= (ithprime(t)+ithprime(t+1))/4; v::integer and isprime(v) end proc:
%p A098016 select(filter, [$1..2000]); # _Robert Israel_, Feb 04 2019
%t A098016 Transpose[Select[Table[{i,Prime[i],Prime[i+1]}, {i,1200}], PrimeQ[Total[Rest[#]]/4]&]][[1]](* _Harvey P. Dale_, Mar 24 2011 *)
%t A098016 Position[Partition[Prime[Range[1100]],2,1],_?(PrimeQ[Total[#]/4]&)]//Flatten (* _Harvey P. Dale_, Sep 11 2022 *)
%o A098016 (PARI) f(n) = for(x=1,n,y=prime(x+1)+prime(x);if(y%4==0 && isprime(y\4),print1(x",")))
%Y A098016 Cf. A000720, A118134.
%K A098016 easy,nonn
%O A098016 1,1
%A A098016 _Cino Hilliard_, Sep 09 2004