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 A282251 Even integers not of the form p + prime(prime(q)) with distinct summands, where p and q are prime. 1
 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 (list; graph; refs; listen; history; text; internal format)
 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 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

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Last modified August 18 12:35 EDT 2018. Contains 313832 sequences. (Running on oeis4.)