A282251 Even integers not of the form p + prime(prime(q)) with distinct summands, where p and q are prime.

2, 4, 6, 10, 20, 26, 32, 56, 80, 86, 116, 122, 152, 176, 214, 218, 248, 332, 382, 422, 446, 556, 586, 596, 620, 634, 904, 928, 1138, 1144, 1180, 1354, 1388, 1390, 1474, 1600, 1684, 3112, 3554, 5128

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

Table of n, a(n) for n=1..40.

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

Cf. A000040, A038580.

Sequence in context: A164141 A034872 A032362 * A176716 A256056 A293281

Adjacent sequences: A282248 A282249 A282250 * A282252 A282253 A282254

Keyword

nonn

Author

Andrei-Lucian Dragoi, Feb 10 2017

Status

approved