A001857 a(1)=2, a(2)=3; for n >= 3, a(n) is smallest number that is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

2, 3, 5, 7, 8, 9, 13, 14, 18, 19, 24, 25, 29, 30, 35, 36, 40, 41, 46, 51, 56, 63, 68, 72, 73, 78, 79, 83, 84, 89, 94, 115, 117, 126, 153, 160, 165, 169, 170, 175, 176, 181, 186, 191, 212, 214, 230, 235, 240, 245, 266, 273, 278, 283, 288, 325, 331, 332, 337, 342

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

An Ulam-type sequence - see A002858 for many further references, comments, etc.

A plot of the first 10^6 terms shows a nearly straight line having a slope of about 11.1. In contrast to A002858, this sequence has many consecutive numbers; of the first 10^6 terms, consecutive numbers appear 141674 times! - T. D. Noe, Jan 21 2008

References

• S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
• R. K. Guy, Unsolved Problems in Number Theory, Section C4.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• S. M. Ulam, Problems in Modern Mathematics, Wiley, NY, 1960, p. ix.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Steven R. Finch, Ulam s-Additive Sequences [From Steven Finch, Apr 20 2019]
• J. Cassaigne and S. R. Finch, A class of 1-additive sequences and additive recurrences
• N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
• Eric Weisstein's World of Mathematics, Ulam Sequence
• Wikipedia, Ulam number
• Index entries for Ulam numbers

Crossrefs

Cf. A100729.

Cf. A199122, A199123.

Programs

Haskell

a001857 n = a001857_list !! (n-1)
a001857_list = 2 : 3 : ulam 2 3 a001857_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Mathematica

s = {2, 3}; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n-s], n/2, 1, 1]] != 2]; n], {100}]; s (* Jean-François Alcover, Sep 08 2011 *)

Extensions

More terms from Jud McCranie