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

2,1

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.

Links

Table of n, a(n) for n=2..23.

Formula

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

Example

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).

Mathematica

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

Crossrefs

Cf. A000040, A001358, A014612, A064653.

Sequence in context: A343682 A343744 A064653 * A079238 A079042 A193455

Adjacent sequences: A130585 A130586 A130587 * A130589 A130590 A130591

Keyword

more,nonn,fini

Author

Jonathan Vos Post, Jun 16 2007

Status

approved