A166081 Natural numbers that are not the sum of two distinct primes.

1, 2, 3, 4, 6, 11, 17, 23, 27, 29, 35, 37, 41, 47, 51, 53, 57, 59, 65, 67, 71, 77, 79, 83, 87, 89, 93, 95, 97, 101, 107, 113, 117, 119, 121, 123, 125, 127, 131, 135, 137, 143, 145, 147, 149, 155, 157, 161, 163, 167, 171, 173, 177, 179, 185, 187, 189, 191, 197, 203

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 06 2009

Keywords

• nonn

Comments

All numbers that appear in A014092 are also in this sequence, by definition.

It seems that, for n > 6, the reverse is also true, however this is unproved. - Ely Golden, Dec 25 2016

All numbers that appear in this sequence but not A014092 must be even semiprimes with no other partitions into primes. - Ely Golden, Dec 25 2016

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 1000 terms from G. C. Greubel)

Crossrefs

Cf. A117929, A006881, A038609 (complement), A014092, A066615.

Programs

Mathematica

Select[Range@ 204, Length@Select[Transpose@{#, Reverse@ # - 1} &@ Range[#] &@ #, Times @@ Boole@ Map[PrimeQ, #] == 1 && First@ # != Last@ # &] == 0 &] (* Michael De Vlieger, Apr 24 2016 *)
max = 1000;
ip = PrimePi[max];
A038609 = Table[Prime[i] + Prime[j], {i, ip}, {j, i + 1, ip}] // Flatten // Union // Select[#, # <= max&]&;
Complement[Range[max], A038609] (* Jean-François Alcover, Mar 24 2020 *)

Formula

{1} U A025584 U A109934. - R. J. Mathar, Oct 08 2009

A000027 \ A038609. - R. J. Mathar, Oct 14 2009