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%I #37 Mar 28 2018 16:06:03
%S 2,4,6,10,20,26,32,56,80,86,116,122,152,176,214,218,248,332,382,422,
%T 446,556,586,596,620,634,904,928,1138,1144,1180,1354,1388,1390,1474,
%U 1600,1684,3112,3554,5128
%N Even integers not of the form p + prime(prime(q)) with distinct summands, where p and q are prime.
%C 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.
%C No other terms up to 10^10.
%C Conjecture 1: This sequence is finite and its largest term is smaller than 2*e^8.
%C 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.
%C 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.
%H A. L. Dragoi, <a href="http://dragoii.com/VBGC_latest_full.pdf">The “Vertical” (generalization of) the Binary Goldbach’s conjecture (VBGC 1.2) as applied on “iterative” primes with (recursive) prime indexes (i-primeths)</a> (2017). DOI: 10.13140/RG.2.2.27963.62245 [<a href="http://dragoii.com/VBGC_latest_extract.pdf">conjecture only</a>]
%e 6 cannot be written as a sum of pair of distincts terms (2_Px, 0_Py).
%e Prime(2) + prime(prime(prime(1))) = 3 + prime(prime(2)) = 3 + prime(3) = 3 + 5 = 8, so 8 is not in this sequence.
%t 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 *)
%t 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 *)
%o (PARI) isokpc(p) = isprime(primepi(p)) && isprime(primepi(primepi(p)));
%o isokpd(p) = isprime(p) && isprime(primepi(p)) && isprime(primepi(primepi(p)));
%o 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
%Y Cf. A000040, A038580.
%K nonn
%O 1,1
%A _Andrei-Lucian Dragoi_, Feb 10 2017
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