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

1,1

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

Links

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

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000720, A118134.

Sequence in context: A177950 A049618 A057292 * A089645 A214259 A287680

Adjacent sequences: A098013 A098014 A098015 * A098017 A098018 A098019

Keyword

easy,nonn

Author

Cino Hilliard, Sep 09 2004

Status

approved