Positive density of Ulam's sequence
Ulam's sequence is defined by $u_1 = 1$, $u_2 = 2$ and, for $n \ge 2$, by letting $u_{n+1}$ be the smallest integer larger than $u_n$ that can be written as $u_i + u_j$ with $i < j \le n$ in exactly one way. It begins $1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, \dots$. Steinerberger presented data suggesting that there is a number $\alpha \approx 2.5714474995$ for which the values $\alpha u_n \bmod 2\pi$ appear to follow a curious limiting distribution; Green asks, as a second question, for an explanation of this.
Problem. Does Ulam's sequence have positive density?
Source: S. M. Ulam, Problems in Modern Mathematics (1964). Listed as Problem 7 in B. Green, 100 open problems (version of January 2026).
Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.
Classification
- Primary
- NT — Number Theory
- Secondary
- CO
- Keywords
- additive number theory, Ulam sequence, unique representations
- Literature checked
- Oct. 26, 2026 by clemens-admin
References
@article{ulam1964siam,
author = {Ulam, S.},
title = {Combinatorial analysis in infinite sets and some physical theories},
journal = {SIAM Review},
volume = {6},
number = {4},
pages = {343--355},
year = {1964},
doi = {10.1137/1006090}
}
@book{ulam1964problems,
author = {Ulam, S. M.},
title = {Problems in Modern Mathematics},
address = {New York},
year = {1964}
}
@article{steinerberger2017,
author = {Steinerberger, Stefan},
title = {A hidden signal in the {U}lam sequence},
journal = {Experimental Mathematics},
volume = {26},
number = {4},
pages = {460--467},
year = {2017},
doi = {10.1080/10586458.2016.1204638},
eprint = {1507.00267},
archivePrefix = {arXiv}
}
@article{finch1992,
author = {Finch, Steven R.},
title = {Are 0-additive sequences always regular?},
journal = {American Mathematical Monthly},
volume = {99},
number = {7},
pages = {671--673},
year = {1992},
doi = {10.1080/00029890.1992.11995911}
}
@article{clement2025steinerberger,
author = {Cl{\'e}ment, Fran{\c{c}}ois and Steinerberger, Stefan},
title = {Small gaps in the {U}lam sequence},
journal = {Comptes Rendus. Math{\'e}matique},
volume = {363},
pages = {941--949},
year = {2025},
doi = {10.5802/crmath.746},
eprint = {2501.16285},
archivePrefix = {arXiv}
}
@misc{erdosproblems342,
author = {Bloom, Thomas F.},
title = {Erd{\H o}s Problem \#342},
howpublished = {\url{https://www.erdosproblems.com/342}},
note = {Accessed 2026-10-26}
}
@misc{green100open,
author = {Green, Ben},
title = {100 open problems},
howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
note = {Problem 7; version of January 2026, most recent update December 2025},
year = {2026}
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00933, "Positive density of Ulam's sequence", clopen.solutions.