A316460 Even integers not of the form prime(prime(x)) + prime(prime(y)) with x > y > 0.

2, 4, 6, 10, 12, 18, 24, 26, 30, 32, 38, 40, 50, 54, 56, 60, 66, 68, 74, 80, 82, 92, 96, 102, 104, 106, 110, 116, 118, 122, 128, 134, 136, 146, 148, 152, 154, 156, 164, 166, 170, 172, 178, 180, 200, 204, 206, 212, 218, 226, 230, 234, 248, 254, 256, 260, 264

Offset

1,1

Comments

No other terms up to 10^10.

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).

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.

Links

Andrei-Lucian Dragoi, Table of n, a(n) for n = 1..771

Andrei-Lucian Dragoi, Program

Andrei-Lucian Dragoi, 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), ResearchGate, 2017.

Example

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.
14 can be written as 14 = 3 + 11 with 3 = 1_P(1) and 11 = 1_P(3), so 14 is not a term.

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000040, A006450.

Sequence arising from the same meta-conjecture: A282251.

Sequence in context: A002491 A045958 A076067 * A065385 A244052 A324059

Adjacent sequences: A316457 A316458 A316459 * A316461 A316462 A316463

Keyword

nonn

Author

Andrei-Lucian Dragoi, Jul 04 2018

Status

approved