A347944 Numbers that cannot be written as the sum of two (not necessarily distinct) terms in A000009.

0, 1, 63, 71, 75, 83, 85, 87, 89, 113, 115, 117, 120, 129, 133, 138, 139, 141, 151, 155, 156, 159, 161, 162, 163, 172, 179, 181, 182, 184, 185, 189, 190, 191, 199, 201, 205, 209, 212, 213, 215, 216, 217, 220, 221, 222, 233, 235, 236, 239, 242, 243, 245, 247, 248, 250

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

(PARI) leng(n) = for(l=0, oo, if(A000009(l)>n-1, return(l))) \\ See A000009 for its program; A000009(0), A000009(1), ..., A000009(l-1) <= n-1
v(n) = my(l=leng(n), v=[]); for(i=0, l-1, v=concat(v, vector(l, j, A000009(i)+A000009(j-1)))); v=vecsort(v); v
list_zero(n) = setminus([0..n], Set(v(n)))

Crossrefs

Cf. A000009, A347943.

Sequence in context: A095579 A095555 A095543 * A073569 A238088 A350960

Adjacent sequences: A347941 A347942 A347943 * A347945 A347946 A347947

Keyword

nonn,easy

Author

Jianing Song, Sep 20 2021

Status

approved