A034797 a(0) = 0; a(n+1) = a(n) + 2^a(n).

0, 1, 3, 11, 2059

Offset: 0

Author

Joseph Shipman (shipman(AT)savera.com)

Keywords

• nonn
• changed

Comments

First impartial game with value n, using natural enumeration of impartial games.

The natural 1-1 correspondence between nonnegative numbers and hereditarily finite sets is given by f(A)=sum over members m of A of 2^f(m). A set can be considered an impartial game where the legal moves are the members. The value of an impartial game is always an ordinal (for finite games, an integer).

The next term, a(5) = 2^2059 + 2059, has 620 decimal digits and is too large to include. - Olivier Gérard, Jun 26 2001

Positions of records in A103318. - N. J. A. Sloane and David Applegate, Mar 21 2005

The first n terms in this sequence form the lexicographically earliest n-vertex clique in the Ackermann-Rado encoding of the Rado graph (an infinite graph in which vertex i is adjacent to vertex j, with iDavid Eppstein, Aug 22 2014

This sequence was used by Spiro to bound the density of refactorable numbers (A033950). - David Eppstein, Aug 22 2014

For any positive integer m, a(1), a(2), ..., a(3^m) modulo 3^m form a complete residue set. - Yifan Xie, Aug 19 2025

References

• J. H. Conway, On Numbers and Games, Academic Press.

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..5
• Wilhelm Ackermann, Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Math. Ann. 114 (1939), no. 1, 305-315.
• David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers, J. Integer Seq. 8 (2005), no. 3, Article 05.3.6, 15 pp. Preprint versions: [pdf, ps].
• O. Kurganskyy and I. Potapov, Reachability problems for PAMs, arXiv preprint arXiv:1510.04121 [cs.NA], 2015.
• Richard Rado, Universal graphs and universal functions, Acta Arith. 9 (1964), 331-340.
• Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory 21 (1985), no. 1, 81-100.
• Wikipedia, Rado graph.

Crossrefs

Cf. A034798, A103318.

Programs

Mathematica

a=0;lst={a};Do[AppendTo[lst,a+=2^a],{n,0,4}];lst (* Vladimir Joseph Stephan Orlovsky, May 06 2010 *)
NestList[#+2^#&,0,5] (* Harvey P. Dale, Mar 22 2020 *)