A064653 -id:A064653 - cp's OEIS frontend

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

Jonathan Vos Post, Jun 16 2007

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.

			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).

Cf. A000040, A001358, A014612, A064653.

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

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

A064915 Positive integers n that are not of the form p + q * a^2, where p and q are primes and a is the smallest prime not dividing n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 22, 24, 26, 28, 30, 36, 42, 48, 54, 60, 66, 72, 77, 84, 90, 96, 102, 108, 114, 120, 126, 174, 180, 210, 240, 270, 300, 330, 420, 630, 840, 1050, 1260

Offset: 1

Dean Hickerson, Oct 13 2001

nonn

There are no other terms up to 10^7. Conjecture: There are no more terms in the sequence.

Cf. A064653.

rep[ n_ ] := Module[ {a, q}, For[ a=2, GCD[ n, a ]!=1, a++, Null ]; For[ q=2, q a^2

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula