OFFSET
1,1
COMMENTS
Define a 0-(prime-index-prime) as a prime, and a k-(prime-index-prime) as a number of the form prime(p) where p is a (k-1)-(prime-index-prime). Then these are the even integers that cannot be expressed as p + q, where p is a 2-(prime-indexed prime), q is a 0-(prime-indexed prime), and p != q.
No other terms up to 10^10.
Conjecture 1: This sequence is finite and its largest term is smaller than 2*e^8.
Conjecture 2: For any m > 0, all even numbers greater than 2*e^(4m) are the sum of a prime and an m-(prime-index-prime). See links.
Conjecture 3: For any m >= 0 and n >= 0, all large enough even numbers are the sum of an m-(prime-index-prime) and an n-(prime-index-prime). See links.
LINKS
A. L. Dragoi, The “Vertical” (generalization of) the Binary Goldbach’s conjecture (VBGC 1.2) as applied on “iterative” primes with (recursive) prime indexes (i-primeths) (2017). DOI: 10.13140/RG.2.2.27963.62245 [conjecture only]
EXAMPLE
6 cannot be written as a sum of pair of distincts terms (2_Px, 0_Py).
Prime(2) + prime(prime(prime(1))) = 3 + prime(prime(2)) = 3 + prime(3) = 3 + 5 = 8, so 8 is not in this sequence.
MATHEMATICA
Function[s, TakeWhile[Select[Complement[Range@ Max@ #, #], EvenQ], # < Max@ s &] &@ Union@ Map[Total, DeleteCases[ Tuples[ {Prime@ Range@ PrimePi@ Max@ s, s}], t_ /; Differences@ t == {0}]]]@ Map[Nest[Prime, #, 2] &, Prime@ Range@ 240] (* Michael De Vlieger, Feb 11 2017 *)
fQ[n_] := Block[{p = 1}, While[q = Prime@ Prime@ Prime@ p; q < n && !PrimeQ[n -q] || 2q == n, p++]; q >= n]; Select[2 Range@ 2600, fQ] (* Robert G. Wilson v, Feb 14 2017 *)
PROG
(PARI) isokpc(p) = isprime(primepi(p)) && isprime(primepi(primepi(p)));
isokpd(p) = isprime(p) && isprime(primepi(p)) && isprime(primepi(primepi(p)));
isok02(n) = forprime(p=2, n, if (p != n-p, if (isokpd(n-p) || (isokpc(p) && isprime(n-p)), return (0)))); 1; \\ Michel Marcus, Feb 10 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei-Lucian Dragoi, Feb 10 2017
STATUS
approved