login
A130588
Integers which are not the sum of a 3-almost prime and a prime.
6
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.
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
KEYWORD
more,nonn,fini
AUTHOR
Jonathan Vos Post, Jun 16 2007
STATUS
approved