A130588 Integers which are not the sum of a 3-almost prime and a prime.

2, 3, 4, 5, 6, 7, 8, 9, 12, 16, 18, 24, 26, 28, 36, 42, 60, 84, 90, 96, 114, 300

Offset: 2

Author

Jonathan Vos Post, Jun 16 2007

Keywords

• more
• nonn
• fini

Comments

T. D. Noe found no more values up to 10000 and agrees with my conjecture that this sequence is probably finite. This is related to Chen's Theorem: "Every 'large' even number may be written as 2n = p + m where p is a prime and m in A001358 is the set of semiprimes (i.e., 2-almost primes)" which itself is related to Goldbach's conjecture. However, we have no proof, merely the sense that it gets easier and easier to find more and more A014612(i) + A000040(j) = n decompositions as n increases.

Examples

			n<10 are in this sequence because the smallest 3-almost prime is 8, hence the smallest 3-almost prime plus prime is 10 = 8 + 2. We have that 282 is not in this sequence because 282 = 125 + 157 = A014612(30) + A000040(37).
		

Crossrefs

Cf. A000040, A001358, A014612, A064653.

Programs

Mathematica

Rest@Complement[Range[1000], Union@Flatten@Outer[Plus, Select[Range[1000], PrimeOmega[#] == 3 &], Prime[Range[PrimePi[1000]]]]] (* Robert Price, Jun 16 2019 *)

Formula

{n such that for no integers i, j is it the case that A014612(i) + A000040(j) = n}.