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

1,1

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

Links

Table of n, a(n) for n=1..56.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A006450, A014092, A166081.

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

Sequence in context: A349080 A337837 A064361 * A064407 A309519 A198392

Adjacent sequences: A303400 A303401 A303402 * A303404 A303405 A303406

Keyword

nonn

Author

Amiram Eldar, May 13 2018

Status

approved