%I #25 Oct 06 2021 03:27:12
%S 0,1,63,71,75,83,85,87,89,113,115,117,120,129,133,138,139,141,151,155,
%T 156,159,161,162,163,172,179,181,182,184,185,189,190,191,199,201,205,
%U 209,212,213,215,216,217,220,221,222,233,235,236,239,242,243,245,247,248,250
%N Numbers that cannot be written as the sum of two (not necessarily distinct) terms in A000009.
%C 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(n1) < N <= A000009(n), at most n^2 numbers among [0,N] are in S (since if N = A000009(m) + A000009(k) then m,k <= n1), so #(S intersect {0,1,...,N})/#{0,1,...,N} <= n^2/(A000009(n1)+2) > 0 as N goes to infinity.
%C This also means that lim_{n>oo} a(n)/n = 1. Proof: Since this is a list we have a(n) >= n1, 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.
%C Note that although 89 is in A000009, it is not the sum of two terms there.
%H Jianing Song, <a href="/A347944/b347944.txt">Table of n, a(n) for n = 1..27999</a> (all terms <= 30000)
%e 63 is a term since it is not the sum of two terms in A000009.
%e 61 is not a term since 61 = 15 + 46.
%e 73 is not a term since 73 = 27 + 46.
%t Select[Range[0,250],!ContainsAny[p,k=1;While[Max[p=PartitionsQ/@Range@k++]<#];#Union@Most@p]&] (* _Giorgos Kalogeropoulos_, Sep 21 2021 *)
%o (PARI) leng(n) = for(l=0, oo, if(A000009(l)>n1, return(l))) \\ See A000009 for its program; A000009(0), A000009(1), ..., A000009(l1) <= n1
%o v(n) = my(l=leng(n), v=[]); for(i=0, l1, v=concat(v, vector(l, j, A000009(i)+A000009(j1)))); v=vecsort(v); v
%o list_zero(n) = setminus([0..n],Set(v(n)))
%Y Cf. A000009, A347943.
%K nonn,easy
%O 1,3
%A _Jianing Song_, Sep 20 2021
