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

2, 3, 9, 11, 23, 32, 54, 58, 67, 76, 86, 103, 164, 188, 200, 202, 208, 210, 243, 311, 351, 354, 374, 414, 420, 427, 441, 468, 515, 539, 559, 588, 621, 639, 650, 652, 662, 670, 693, 708, 748, 752, 769, 811, 816, 823, 842, 883, 889, 939, 943, 963, 970, 1006, 1009

Offset: 1

Author

Cino Hilliard, Sep 09 2004

Keywords

• easy
• nonn

Comments

Conjecture: (1/2)(prime(x+1) + prime(x)) is not prime for all x.

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

Examples

			Prime(2+1) + prime(2) = 5+3 = 8. 1/4(8) = 2. 2 is the first entry.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000720, A118134.

Programs

Maple

filter:= proc(t) local v; v:= (ithprime(t)+ithprime(t+1))/4; v::integer and isprime(v) end proc:
select(filter, [$1..2000]); # Robert Israel, Feb 04 2019

Mathematica

Transpose[Select[Table[{i,Prime[i],Prime[i+1]}, {i,1200}], PrimeQ[Total[Rest[#]]/4]&]][[1]](* Harvey P. Dale, Mar 24 2011 *)
Position[Partition[Prime[Range[1100]],2,1],?(PrimeQ[Total[#]/4]&)]//Flatten (* _Harvey P. Dale, Sep 11 2022 *)

PARI

f(n) = for(x=1,n,y=prime(x+1)+prime(x);if(y%4==0 && isprime(y\4),print1(x",")))

Formula

a(n) = A000720(2*A118134(n)-1). - Robert Israel, Feb 04 2019