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%I #41 Aug 03 2018 00:57:50
%S 2,4,6,10,12,18,24,26,30,32,38,40,50,54,56,60,66,68,74,80,82,92,96,
%T 102,104,106,110,116,118,122,128,134,136,146,148,152,154,156,164,166,
%U 170,172,178,180,200,204,206,212,218,226,230,234,248,254,256,260,264
%N Even integers not of the form prime(prime(x)) + prime(prime(y)) with x > y > 0.
%C No other terms up to 10^10.
%C Define a 1-(prime-index-prime) with index x to be a number of the form prime(prime(x)). These are the even integers that cannot be expressed as 1_P(x) + 1_P(y), with 1_P(x) != 1_P(y).
%C Conjecture: Define an m-(prime-index-prime) as having "m" prime-on-prime iterations, For any m >= 0 and n >= 0, all sufficiently large even numbers are the sum of an m-(prime-index-prime) and an n-(prime-index-prime). See links.
%H Andrei-Lucian Dragoi, <a href="/A316460/b316460.txt">Table of n, a(n) for n = 1..771</a>
%H Andrei-Lucian Dragoi, <a href="https://drive.google.com/drive/u/1/folders/0Bws5l5MW9z0pY0lpbWJRZmNaMGc">Program</a>
%H Andrei-Lucian Dragoi, <a href="https://dx.doi.org/10.13140/RG.2.2.14014.28484">The "Vertical" (generalization of) the Binary Goldbach's conjecture (VBGC 1.5e) as applied on "iterative" primes with (recursive) prime indexes (i-primeths) (the conjecture only)</a>, ResearchGate, 2017.
%e 6 cannot be written as a sum of pair of distinct numbers (1_P(x), 1_P(y)), as 6 = 3 + 3 is the only way to write 6 as the sum of two primes, so 6 is a term.
%e 14 can be written as 14 = 3 + 11 with 3 = 1_P(1) and 11 = 1_P(3), so 14 is not a term.
%t Complement[2 Range[(Prime[Prime[998]] + 1)/2], Sort@ Flatten@ Table[ Prime[Prime[x]] + Prime[Prime[y]], {y, 2, 998}, {x, y - 1}]] (* _Robert G. Wilson v_, Aug 01 2018 *)
%o (PARI) is(n) = if(n%2==1, return(0)); for(x=2, n, for(y=1, x-1, if(n==prime(prime(x)) + prime(prime(y)), return(0)))); 1 \\ _Felix Fröhlich_, Jul 06 2018
%Y Cf. A000040, A006450.
%Y Sequence arising from the same meta-conjecture: A282251.
%K nonn
%O 1,1
%A _Andrei-Lucian Dragoi_, Jul 04 2018
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