OFFSET
1,3
COMMENTS
Most natural numbers are here: this sequence has natural density 1. Proof: Let S be the set of numbers can be written as the sum of two terms in A000009. For A000009(n-1) < N <= A000009(n), at most n^2 numbers among [0,N] are in S (since if N = A000009(m) + A000009(k) then m,k <= n-1), so #(S intersect {0,1,...,N})/#{0,1,...,N} <= n^2/(A000009(n-1)+2) -> 0 as N goes to infinity.
This also means that lim_{n->oo} a(n)/n = 1. Proof: Since this is a list we have a(n) >= n-1, so liminf_{n->oo} a(n)/n >= 1. If limsup_{n->oo} a(n)/n > 1, there exists eps > 0 and n_1 < n_2 < ... < n_i < ... such that a(n_i)/(n_i) > 1+eps for all i, then #({0,1,...,a(n_i)}\S)/#{0,1,...,a(n_i)} = #{a(1),a(2),...,a(n_i)}/#{0,1,...,a(n_i)} = (n_i)/(a(n_i)+1) < 1/(1+eps) for all i, contradicting with the fact that this sequence has natural density 1. Hence limsup_{n->oo} a(n)/n <= 1, so lim_{n->oo} a(n)/n = 1.
Note that although 89 is in A000009, it is not the sum of two terms there.
LINKS
Jianing Song, Table of n, a(n) for n = 1..27999 (all terms <= 30000)
EXAMPLE
63 is a term since it is not the sum of two terms in A000009.
61 is not a term since 61 = 15 + 46.
73 is not a term since 73 = 27 + 46.
MATHEMATICA
Select[Range[0, 250], !ContainsAny[p, k=1; While[Max[p=PartitionsQ/@Range@k++]<#]; #-Union@Most@p]&] (* Giorgos Kalogeropoulos, Sep 21 2021 *)
PROG
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jianing Song, Sep 20 2021
STATUS
approved