A303403 Even numbers that are not the sum of two prime-indexed primes.

2, 4, 12, 18, 24, 26, 30, 32, 38, 40, 50, 54, 56, 60, 66, 68, 74, 80, 92, 96, 102, 104, 106, 110, 116, 122, 128, 136, 146, 148, 152, 154, 156, 164, 170, 172, 178, 180, 200, 204, 206, 212, 226, 230, 234, 248, 256, 260, 264, 268, 276, 290, 292, 296, 298, 302

Offset: 1

Author

Amiram Eldar, May 13 2018

Keywords

• nonn

Comments

Bayless et al. conjectured that every even number larger than 80612 is the sum of two prime-indexed primes. If the conjecture is true then this sequence is finite with 733 terms.

Similarly, it appears that 322704332 is the largest of the 1578727 even numbers that cannot be written as prime(prime(prime(i))) + prime(prime(prime(j))). - Giovanni Resta, May 31 2018

Examples

			20 is not in the sequence since 20 = 17 + 3 = prime(7) + prime(2).  2 and 7 are primes, so 3 and 17 are prime-indexed primes. - _Michael B. Porter_, May 21 2018
		

Links

• Jonathan Bayless, Dominic Klyve, and Tomás Oliveira e Silva, New Bounds and Computations on Prime-Indexed Primes, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 13, Paper A43, 2013.

Crossrefs

Cf. A006450, A014092, A166081.

Equals 2*A174682. - Michel Marcus, May 18 2018

Programs

Mathematica

pipQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]; s1falsifiziertQ[s_]:= Module[{ip=IntegerPartitions[s, {2}], widerlegt=False}, Do[If[pipQ[ip[[i, 1]] ] ~And~ pipQ [ip[[i, 2]] ], widerlegt = True; Break[]], {i, 1, Length[ip]}]; widerlegt]; Select[Range[2500],EvenQ[#]&& s1falsifiziertQ[ # ]==False&] (* after Michael Taktikos at A014092 *)
(* or *) p = Prime@ Prime@ Range@ PrimePi@ PrimePi@ 302; Select[Range[2, 302, 2], IntegerPartitions[#, {2}, p] == {} &] (* Giovanni Resta, May 31 2018 *)

PARI

isok(n) = {if (n % 2, return (0)); forprime(p=2, n/2, if (isprime(primepi(p)) && isprime(n-p) && isprime(primepi(n-p)), return (0));); return (1);} \\ Michel Marcus, May 18 2018