1,2

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.

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

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]

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).

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

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.

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)

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.16.2.

R. K. Guy, Unsolved Problems in Number Theory, C4.

D. E. Knuth, The Art of Computer Programming, Volume 4A, Section 7.1.3.

Popular Computing (Calabasas, CA), "Sieves", Vol. 2 (No. 13, Apr 1974), pp. 6-7.

R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.

B. Recaman, Questions on a sequence of Ulam, Amer. Math. Monthly, 80 (1973), 919-920.

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

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. Ulam, Combinatorial analysis in infinite sets and some physical theories. SIAM Rev. 6 1964 343-355.

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. 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.

Jud McCranie, Table of n, a(n) for n = 1..10000

Richard A. Becker, Plot of residuals a(n) - 13.5167*n for n <= 3000000 [uses Jud McCranie's values of a(n)].

S. R. Finch, Ulam s-Additive Sequences

S. R. Finch, Stolarsky-Harborth Constant

D. E. Knuth, Downloadable programs

Ed Pegg, Jr., Graph of 10^6 terms of a(n) - 13.5*n

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

David W. Wilson, Plot of initial terms, 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.

Index entries for Ulam numbers

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]

Cf. A054540, A072832, A002859, A003667, A001857, A007300, A117140.

Cf. A080287, A080288.

Sequence in context: A033056 A060469 A080329 this_sequence A105799 A102463 A056829

Adjacent sequences: A002855 A002856 A002857 this_sequence A002859 A002860 A002861

nonn,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Jud McCranie (j.mccranie(AT)comcast.net)