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%I A002858 M0557 N0201
%S A002858 1,2,3,4,6,8,11,13,16,18,26,28,36,38,47,48,53,57,62,69,72,77,82,87,
%T A002858 97,99,102,106,114,126,131,138,145,148,155,175,177,180,182,189,197,206,
%U A002858 209,219,221,236,238,241,243,253,258,260,273,282,309,316,319,324,339
%N A002858 Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) 
               which is a unique sum of two distinct earlier terms.
%C A002858 Ulam conjectured that this sequence has density 0. However, calculations 
               up to 4*10^7 (Jud McCranie) indicate that the density hovers near 
               0.074.
%C A002858 A plot of the first 3 million terms shows that they lie very close to 
               the straight line 13.51*n, so even if we cannot prove it, we believe 
               we now know how this sequence grows (see the plots in the links below). 
               - N. J. A. Sloane (njas(AT)research.att.com), Sep 27 2006
%C A002858 After a few initial terms, the sequence settles into a regular pattern 
               of dense clumps separated by sparse gaps, with period 21.601584+. 
               This pattern continues up to at least a(n) = 5*10^6. (This comment 
               is just a qualitative statement about the wavelike distribution of 
               Ulam numbers, not meant to imply that every period includes Ulam 
               numbers.) [David Wilson]
%C A002858 D. E. Knuth (Sep 26 2006) remarks that a(4952)=64420 and a(4953)=64682 
               (a gap of more than ten "dense clumps"); and there is a gap of 315 
               between a(18857) and a(18858).
%C A002858 1,2,3,47 are the only values of x < 40000000 such that x and x+1 are 
               both Ulam numbers - Jud McCranie, Jun 08, 2001
%C A002858 Comments from Jud McCranie on David Wilson's illustration, Jun 20 2008: 
               (Start) The integers are shown from left to right, top to bottom, 
               with a dot where there is an Ulam number. I think his plot is 216 
               wide. The local density of Ulam numbers goes in waves with a period 
               of 21.6+, so his plot shows ten cycles.
%C A002858 When they are arranged that way you can see the waves. The crests of 
               the density waves don't always have Ulam numbers there but the troughs 
               are practically void of Ulam numbers. I noticed that the ratio of 
               that period (21.6+) to the frequency of Ulam numbers (1 in 13.52) 
               is very close to 8/5. (End)
%D A002858 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.2.
%D A002858 R. K. Guy, Unsolved Problems in Number Theory, C4.
%D A002858 D. E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.3.
%D A002858 Popular Computing (Calabasas, CA), "Sieves", Vol. 2 (No. 13, Apr 1974), 
               pp. 6-7.
%D A002858 R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 
               31-71.
%D A002858 B. Recaman, Questions on a sequence of Ulam, Amer. Math. Monthly, 80 
               (1973), 919-920.
%D A002858 J. Schmerl and E. Spiegel, The regularity of some 1-additive sequences. 
               J. Combin. Theory Ser. A 66 (1994), no. 1, 172-175. Math. Rev. 95h:11010
%D A002858 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A002858 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A002858 S. Ulam, Combinatorial analysis in infinite sets and some physical theories. 
               SIAM Rev. 6 1964 343-355.
%D A002858 M. C. Wunderlich, The improbable behavior of Ulam's summation sequence, 
               pp. 249-257 of A. O. L. Atkin and B. J. Birch, editors, Computers 
               in Number Theory. Academic Press, NY, 1971.
%D A002858 D. Zeitlin, Ulam's sequence {U_n}, U_1=1, U_2=2, is a complete sequence, 
               Notices Amer. Math. Soc., 22 (No. 7, 1975), Abstract 75T-A267, p. 
               A-707.
%H A002858 Jud McCranie, <a href="b002858.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A002858 Richard A. Becker, <a href="a002858.ps">Plot of residuals a(n) - 13.5167*n 
               for n <= 3000000</a> [uses Jud McCranie's values of a(n)].
%H A002858 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/stlrsky/sadd/
               sadd.html">Ulam s-Additive Sequences</a>
%H A002858 S. R. Finch, <a href="http://algo.inria.fr/bsolve/constant/stlrsky/stlrsky.html">
               Stolarsky-Harborth Constant</a>
%H A002858 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs.html">
               Downloadable programs</a>
%H A002858 Ed Pegg, Jr., <a href="http://www.wolframscience.com/preview/nks_pages/
               ?NKS0908.gif">Graph of 10^6 terms of a(n) - 13.5*n</a>
%H A002858 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               UlamSequence.html">Link to a section of The World of Mathematics.</
               a>
%H A002858 David W. Wilson, <a href="a002858.png">Plot of initial terms</a>, showing 
               their quasiperiodicity as vertical bars. The image width was chosen 
               to include approximately 10 periods. For an explanation of this picture, 
               see Comments above.
%H A002858 <a href="Sindx_U.html#Ulam_num">Index entries for Ulam numbers</a>
%t A002858 Ulam4Compiled = Compile[{{nmax, _Integer}, {init, _Integer, 1}, {s, _Integer}}, 
               Module[{ulamhash = Table[0, {nmax}], ulam = init}, ulamhash[[ulam]] 
               = 1; Do[ If[Quotient[Plus @@ ulamhash[[i - ulam]], 2] == s, AppendTo[ulam, 
               i]; ulamhash[[i]] = 1], {i, Last[init] + 1, nmax}]; ulam]]; Ulam4Compiled[355, 
               {1, 2}, 1]
%Y A002858 Cf. A054540, A072832, A002859, A003667, A001857, A007300, A117140.
%Y A002858 Cf. A080287, A080288.
%Y A002858 Sequence in context: A033056 A060469 A080329 this_sequence A105799 A102463 
               A056829
%Y A002858 Adjacent sequences: A002855 A002856 A002857 this_sequence A002859 A002860 
               A002861
%K A002858 nonn,nice
%O A002858 1,2
%A A002858 N. J. A. Sloane (njas(AT)research.att.com).
%E A002858 More terms from Jud McCranie (j.mccranie(AT)comcast.net)

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