A348413 a(0) = A002858(1) = 1, followed by the greatest Ulam numbers A002858 to form a complete sequence (see algorithm below).

1, 2, 4, 8, 16, 28, 57, 114, 221, 451, 893, 1792, 3549, 7104, 14212, 28445, 56894, 113792, 227554, 455124, 910208, 1820449, 3640907, 7281813, 14563613, 29127251, 58254501, 116508984, 233017889, 466035877, 932071736

Offset: 0

Author

Frank M Jackson, Oct 17 2021

Keywords

• nonn
• more

Comments

This sequence starts at a(0)=1, subsequent terms a(n) for n>0 being obtained by selecting the (greatest Ulam number) <= 1+Sum_{i=0..n-1} a(i). This ensures that the sequence is complete because Sum_{i=0..n-1} a(i) >= a(n)-1, for all n>=0 and a(0)=1, is a necessary and sufficient condition for completeness.

Examples

			Given that the first 7 terms of the sequence are 1, 2, ..., 28, 57 then a(7)=(greatest Ulam number) <= (1+2+...+28, 57) + 1 = 117, hence a(7)=114.
		

Links

• Wikipedia, Ulam number.

Crossrefs

Cf. A002858.

Programs

Mathematica

lst1 = Last/@ReadList["https://oeis.org/A002858/b002858.txt", {Number, Number}]; lst={1, 2}; n=3; Do[s=Total@lst; While[s+1>=lst1[[n]], n++]; AppendTo[lst, lst1[[n-1]]], 16]; lst

Formula

a(n) = (greatest Ulam number) <= 1+Sum_{i=0..n-1} a(i), with a(0) = 1.

Extensions

a(18)-a(30) from Amiram Eldar, Oct 17 2021