OFFSET

1,2

COMMENTS

Ulam conjectured that this sequence has density 0. However, calculations up to 6.759*10^8 (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, 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 W. Wilson

Don 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 < 6.759*10^8 such that x and x+1 are both Ulam numbers. - Jud McCranie, Jun 08 2001. This holds through the first 28 billion Ulam numbers - Jud McCranie, Jan 07 2016.

From Jud McCranie on David W. 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)

a(50000000) = 675904508. - Jud McCranie, Feb 29 2012

a(100000000) = 1351856726. - Jud McCranie, Jul 31 2012

a(1000000000) = 13517664323. - Jud McCranie, Aug 28 2015

a(28000000000) = 378485625853 - Philip Gibbs & Jud McCranie, Sep 09 2015

3 (=1+2) and 131 (=62+69) are the only two Ulam numbers in the first 28 billion Ulam numbers that are the sum of two consecutive Ulam numbers. - Jud McCranie, Jan 09 2016

Named after the Polish-American scientist Stanislaw Ulam (1909-1984). - Amiram Eldar, Jun 08 2021

REFERENCES

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

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

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

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).

Marvin 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.

David 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.

LINKS

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, postscript file [uses Jud McCranie's values of a(n)].

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

Thomas Bloom, Problem 342, Erdős Problems.

Henning Fernau and Kshitij Gajjar, The Space Complexity of Sum Labelling, arXiv:2107.12973 [cs.DS], 2021.

Steven R. Finch, Ulam s-Additive Sequences. [From Steven Finch, Apr 20 2019]

Steven R. Finch, Stolarsky-Harborth Constant.

Steven R. Finch, Stolarsky-Harborth Constant. [From the Wayback machine]

Philip Gibbs, An Efficient Way to Compute Ulam Numbers.

Philip Gibbs, A Conjecture for Ulam Sequences.

Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).

Alois P. Heinz, Ulam spiral of Ulam numbers.

Donald E. Knuth, Downloadable programs

Noah Kravitz and Stefan Steinerberger, Ulam Sequences and Ulam Sets, arXiv:1705.01883 [math.CO], 2017.

Borys Kuca, Structures in Additive Sequences, arXiv:1804.09594 [math.NT], 2018. See p. 3.

J. W. Moon, R. K. Guy, and N. J. A. Sloane, Correspondence, 1973.

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

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #8. [Annotated and scanned copy]

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

Bernardo Recamán, Questions on a sequence of Ulam, Amer. Math. Monthly, Vol. 80, No. 8 (1973), pp. 919-920.

Ivano Salvo and Agnese Pacifico, Computing Integer Sequences: Filtering vs Generation (Functional Pearl), arXiv:1807.11792 [cs.PL], 2018.

James Schmerl and Eugene Spiegel, The regularity of some 1-additive sequences, J. Combin. Theory. Ser. A, Vol. 66, No. 1 (1994), pp. 172-175. Math. Rev. 95h:11010.

Arseniy Sheydvasser, The Ulam Sequence of the Integer Polynomial Ring, arXiv:1910.00109 [math.NT], 2019.

N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).

Arseniy (Senia) Sheydvasser, The Ulam Sequence of Linear Integer Polynomials, J. Int. Seq., Vol. 24 (2021), Article 21.10.8.

Stefan Steinerberger, A hidden signal in the Ulam sequence, Research Report YALEU/DCS/TR-1508, Yale University, 2015.

Stefan Steinerberger, A hidden signal in the Ulam sequence, Figure 2 from Research Report YALEU/DCS/TR-1508, Yale University, 2015.

Stefan Steinerberger, The Ulam Sequence, blog post, April 12, 2016.

Stanislaw M. Ulam, On some mathematical problems connected with patterns of growth of figures, pp. 215-224 of R. E. Bellman, ed., Mathematical Problems in the Biological Sciences, Proc. Sympos. Applied Math., Vol. 14, Amer. Math. Soc., 1962 [Annotated scanned copy]

Stanislaw Ulam, Combinatorial analysis in infinite sets and some physical theories, SIAM Rev., Vol. 6, No. 4 (1964), pp. 343-355. Reprinted in Selected Papers, MIT Press, see p. 393.

Eric Weisstein's World of Mathematics, Ulam Sequence.

Wikipedia, Ulam number.

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.

Robert G. Wilson v, Letter to N. J. A. Sloane, Sep. 1992.

David 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. (Annotated scanned copy)

MAPLE

UlamList := proc(len) local isUlam, nextUlam, behead; behead := u -> u[2..numelems(u)]; isUlam := proc(n, h, u, r) local hu, tu, hr, tr; hu := u[1]; hr := r[1]; if h = 2 then return false fi; if hr <= hu then return evalb(h = 1) fi; if (hr + hu) = n then tu := behead(u); tr := behead(r); return isUlam(n, h+1, tu, tr) fi; if (hr + hu) < n then tu := behead(u): return isUlam(n, h, tu, r) fi; tr := behead(r); isUlam(n, h, u, tr) end: nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len-1 then return [op(u), n] fi; nextUlam(n+1, [op(u), n], [n, op(r)]) else nextUlam(n+1, u, r) fi end: nextUlam(3, [1, 2], [2, 1]) end:

UlamList(59); # Peter Luschny, Apr 05 2019

MATHEMATICA

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]]; ulams = Ulam4Compiled[355, {1, 2}, 1]

(* Second program: *)

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

findUlams[s_List, j_Integer] := Block[{k = s[[-1]] + 1, ss = Plus @@@ Subsets[s, {j}]}, While[ Count[ss, k] != 1, k++]; Append[s, k]]; ulams = Nest[findUlams[#, 2] &, {1, 2}, 70] (* Robert G. Wilson v, Jul 05 2014 *)

PROG

(Haskell)

a002858 n = a002858_list !! (n-1)

a002858_list = 1 : 2 : ulam 2 2 a002858_list

ulam :: Int -> Integer -> [Integer] -> [Integer]

ulam n u us = u' : ulam (n + 1) u' us where

u' = f 0 (u+1) us'

f 2 z _ = f 0 (z + 1) us'

f e z (v:vs) | z - v <= v = if e == 1 then z else f 0 (z + 1) us'

| z - v `elem` us' = f (e + 1) z vs

| otherwise = f e z vs

us' = take n us

-- Reinhard Zumkeller, Nov 03 2011

(Python)

def isUlam(n, h, u, r):

if h == 2: return False

hu = u[0]; hr = r[0]

if hr <= hu: return h == 1

if hr + hu > n: r = r[1:]

elif hr + hu < n: u = u[1:]

else: h += 1; r = r[1:]; u = u[1:]

return isUlam(n, h, u, r)

def UlamList(length):

u = [1, 2]; r = [2, 1]; n = 2

while len(u) < length:

n += 1

if isUlam(n, 0, u[:], r[:]):

u.append(n); r.insert(0, n)

return u

print(UlamList(59)) # Peter Luschny, Apr 06 2019

(Julia)

function isUlam(u, n, h, i, r)

h == 2 && return false

ur = u[r]; ui = u[i]

ur <= ui && return h == 1

if ur + ui > n

r -= 1

elseif ur + ui < n

i += 1

else

h += 1; i += 1; r -= 1

end

isUlam(u, n, h, i, r)

end

function UlamList(len)

u = Array{Int, 1}(undef, len)

u[1] = 1; u[2] = 2; i = 2; n = 2

while i < len

n += 1

if isUlam(u, n, 0, 1, i)

i += 1

u[i] = n

end

end

return u

end

println(UlamList(59)) # Peter Luschny, Apr 07 2019

(PARI) aupto(N)= my(seen=vector(N), U=[]); seen[1]=seen[2]=1; for(i=1, N, if(1==seen[i], for(j=1, #U, my(sum=i+U[j]); if(sum>N, break); seen[sum]++); U=concat(U, i))); U \\ Ruud H.G. van Tol, Dec 29 2022

CROSSREFS

KEYWORD

nonn,nice

AUTHOR

EXTENSIONS

More terms from Jud McCranie

STATUS

approved