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%I #7 Jun 16 2019 22:30:10
%S 2,3,4,5,6,7,8,9,12,16,18,24,26,28,36,42,60,84,90,96,114,300
%N Integers which are not the sum of a 3-almost prime and a prime.
%C 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.
%F {n such that for no integers i, j is it the case that A014612(i) + A000040(j) = n}.
%e 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).
%t Rest@Complement[Range[1000], Union@Flatten@Outer[Plus, Select[Range[1000], PrimeOmega[#] == 3 &], Prime[Range[PrimePi[1000]]]]] (* _Robert Price_, Jun 16 2019 *)
%Y Cf. A000040, A001358, A014612, A064653.
%K more,nonn,fini
%O 2,1
%A _Jonathan Vos Post_, Jun 16 2007
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