A002858 -id:A002858 - cp's OEIS frontend

A214603 Location of the first gap of exactly n in Ulam numbers, or zero if none is known. The zero terms are conjectural.

1, 4, 8, 53, 48, 0, 62, 18, 38, 87, 0, 114, 260, 0, 221, 0, 568, 0, 627, 155, 0, 646, 0, 1724, 695, 0, 282, 0, 1433, 751, 0, 36460, 0, 949, 483379914, 0, 8018, 0, 1257, 0, 1553, 1858, 0, 4057, 0, 9262, 3825, 0, 10606, 0, 115835, 3153, 0, 0, 0, 6982

Offset: 1

Jud McCranie, Jul 22 2012

nonn

Some terms are not believed to exist. However, the first gap of 35 occurs at 483379914 and the first gap of 88 occurs at 1127544124. - Jud McCranie, Jul 31 2012

			The first gap of 5 in Ulam numbers occurs after 48, so a(5) = 48.

Jud McCranie, Known terms up to 890

Cf. A002858, A080287, A080288.

A307331 The number of Ulam numbers below 10^n.

Original entry on oeis.org

6, 26, 125, 827, 7584, 74083, 740368, 7399353, 73976842, 739778038, 7397814233, 73979274540

Offset: 1

Amiram Eldar, Apr 02 2019

The terms a(1)-a(12) are from the paper by Gibbs and McCranie.
The number of Ulam numbers not exceeding 10^n can be calculated from this sequence using the fact that up to 10^12 the only powers of 10 that are Ulam numbers are 10^6 and 10^11.
The asymptotic density D of Ulam numbers is known to be 0 <= D <= 0.4. Gibbs and McCranie estimated it from this sequence as ~ 0.073979.

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

Cf. A002858.

A378795 Ulam numbers that are semiprimes.

Original entry on oeis.org

4, 6, 26, 38, 57, 62, 69, 77, 82, 87, 106, 145, 155, 177, 206, 209, 219, 221, 253, 309, 319, 339, 341, 358, 382, 451, 485, 497, 502, 566, 685, 695, 734, 781, 849, 866, 893, 905, 949, 1018, 1037, 1079, 1081, 1101, 1157, 1167, 1169, 1186, 1191, 1257, 1313, 1355

Offset: 1

Massimo Kofler, Dec 07 2024

nonn

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.

			4 is a term because 4=2*2 is the product of 2 (not distinct) primes and 4 is an Ulam number.
6 is a term because 6=2*3 is the product of 2 distinct primes and 6 is an Ulam number.
57 is a term because 57=3*19 is the product of 2 distinct primes and 57 is an Ulam number.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A001358 and A002858.

Cf. A068820.

N:= 5000: # for terms <= N
U:= [1, 2]: V:= Vector(N): V[3]:= 1: R:= NULL: count:= 0:
for i from 3 do
   for k from U[-1]+1 to N do
     if V[k] = 1 then
       J:= select(`<=`, U +~ k, N);
       V[J]:= V[J] +~ 1;
       U:= [op(U), k];
       if numtheory:-bigomega(k) = 2 then R:= R, k; count:= count+1;  fi;
       break
     fi
   od;
   if k > N then break fi;
od:
R; # Robert Israel, Jan 24 2025

seq[numUlams_] := Module[{ulams = {1, 2}}, Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[DeleteCases[Intersection[ulams, n - ulams], n/2, 1, 1]] != 2]; n], {numUlams}]; Select[ulams, PrimeOmega[#] == 2 &]]; seq[200] (* Amiram Eldar, Dec 07 2024, after Jean-François Alcover at A002858 *)

A006844 a(1)=4, a(2)=5; thereafter a(n) is smallest number that is greater than a(n-1) and having a unique representation as a(j) + a(k) for j

Original entry on oeis.org

4, 5, 9, 13, 14, 17, 19, 21, 24, 25, 27, 35, 37, 43, 45, 47, 57, 67, 69, 73, 77, 83, 93, 101, 105, 109, 113, 115, 123, 125, 133, 149, 153, 163, 173, 197, 201, 205, 209, 211, 213, 217, 219, 227, 229, 235, 237, 239

Offset: 1

N. J. A. Sloane

This is the 1-additive sequence with base {4,5}. Apart from three extra terms (4, 14, 24) in the initial segment, this breaks up naturally into segments of 32 terms each. [Finch, 1992]. - N. J. A. Sloane, Aug 12 2015

An Ulam-type sequence - see A002858 for many further references, comments, etc. - T. D. Noe, Jan 21 2008

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
R. K. Guy, "s-Additive sequences," preprint, 1994.
R. K. Guy, Unsolved Problems in Number Theory, Section C4.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..1000
Experimental Mathematics, Home Page
Steven R. Finch, Stolarsky-Harborth Constant [Broken link]
Steven R. Finch, Stolarsky-Harborth Constant [From the Wayback machine]
Steven R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
R. Queneau, Sur les suites s-additives, J. Combin. Theory, A12 (1972), 31-71.
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

a006844 n = a006844_list !! (n-1)
a006844_list = 4 : 5 : ulam 2 5 a006844_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

s = {4, 5}; n0 = 9; dn = 32; m = 192; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n - s], n/2, 1, 1]] != 2]; n], {n0 + dn}]; Clear[a]; a[n_] := a[n] = If[n <= n0 + dn, s[[n]], a[n - dn] + m]; Table[a[n], {n, 1, 200}] (* Jean-François Alcover, Apr 03 2013 *)

For n>9, a(n+32) = a(n) + 192. - T. D. Noe, Jan 21 2008

A048951 (2,4) Ulam sequence.

Original entry on oeis.org

2, 4, 6, 8, 12, 16, 22, 26, 32, 36, 52, 56, 72, 76, 94, 96, 106, 114, 124, 138, 144, 154, 164, 174, 194, 198, 204, 212, 228, 252, 262, 276, 290, 296, 310, 350, 354, 360, 364, 378, 394, 412, 418, 438, 442, 472, 476, 482, 486, 506, 516, 520, 546, 564, 618, 632

Offset: 1

Eric W. Weisstein

nonn

This sequence is 2*A002858, since in general U(k*m, k*n) = k * U(m, n), where U(m, n) is the (m, n)-Ulam sequence. - Enrique Navarrete, May 05 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A002858.

a048951 n = a048951_list !! (n-1)
a048951_list = 2 : 4 : ulam 2 4 a048951_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

A049821 a(n) = j + k, where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Original entry on oeis.org

3, 4, 5, 6, 8, 9, 12, 12, 14, 15, 15, 17, 18, 18, 20, 20, 22, 23, 23, 25, 25, 27, 28, 28, 35, 28, 29, 35, 33, 38, 35, 41, 37, 37, 39, 41, 46, 48, 43, 51, 45, 53, 48, 48, 50, 50, 58, 52, 60, 54, 56, 62, 56, 65, 59, 61, 61, 63, 70, 64, 66, 71, 66, 73, 69, 77, 71, 79, 73, 83, 74, 76, 78

Offset: 3

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Nov 20 2019: (Start)
A002859(3) = 4 = 1 + 3 = A002859(1) + A002859(2), so a(3) = 1 + 2 = 3.
A002859(4) = 5 = 1 + 4 = A002859(1) + A002859(3), so a(4) = 1 + 3 = 4.
A002859(5) = 6 = 1 + 5 = A002859(1) + A002859(4), so a(5) = 1 + 4 = 5.
A002859(6) = 8 = 3 + 5 = A002859(2) + A002859(4), so a(6) = 2 + 4 = 6.
A002859(7) = 10 = 4 + 6 = A002859(3) + A002859(5), so a(7) = 3 + 5 = 8.
(End)

Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.

Cf. A002858, A002859, A049877, A049878, A049879.

# First we modify Peter Luschny's program from A002858 (with len >= 3):
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; end if; if hr <= hu then return evalb(h = 1); end if; if hr + hu = n then tu := behead(u); tr := behead(r); return isUlam(n, h + 1, tu, tr); end if; if hr + hu < n then tu := behead(u); return isUlam(n, h, tu, r); end if; tr := behead(r); isUlam(n, h, u, tr); end proc; nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len - 1 then return [op(u), n]; end if; nextUlam(n + 1, [op(u), n], [n, op(r)]); else nextUlam(n + 1, u, r); end if; end proc; nextUlam(3, [1, 3], [3, 1]); end proc:
# Next we create a function to calculate a(n) for given n >= 3:
a := proc(n) local u, a, i, j: u := 0: if 3 <= n then a := UlamList(n): for i to n - 2 do for j from i + 1 to n - 1 do if a[n] = a[i] + a[j] then u := i+j: end if: end do: end do: end if: u: end proc:
# Finally, we create a list of values for a(n):
seq(a(n), n=3..100); # Petros Hadjicostas, Nov 20 2019

Name edited by and typo in the data corrected by Petros Hadjicostas, Nov 20 2019

More terms from Petros Hadjicostas, Nov 20 2019

A049877 a(n) = max(j,k), where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Original entry on oeis.org

2, 3, 4, 4, 5, 6, 8, 9, 9, 11, 12, 12, 14, 15, 16, 17, 17, 19, 20, 21, 22, 22, 24, 25, 20, 27, 27, 27, 30, 27, 32, 27, 34, 35, 36, 36, 27, 27, 40, 27, 42, 36, 44, 45, 46, 47, 31, 49, 33, 51, 52, 35, 53, 38, 56, 57, 58, 58, 43, 61, 62, 44, 63, 46, 66, 50, 68, 52, 70, 47, 72, 73, 73, 60

Offset: 3

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Nov 20 2019: (Start)
A002859(3) = 4 = 1 + 3 = A002859(1) + A002859(2), so a(3) = max(1,2) = 2.
A002859(4) = 5 = 1 + 4 = A002859(1) + A002859(3), so a(4) = max(1,3) = 3.
A002859(5) = 6 = 1 + 5 = A002859(1) + A002859(4), so a(5) = max(1,4) = 4.
A002859(6) = 8 = 3 + 5 = A002859(2) + A002859(4), so a(6) = max(2,4) = 4.
A002859(7) = 10 = 4 + 6 = A002859(3) + A002859(5), so a(7) = max(3,5) = 5.
(End)

Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.

Cf. A002858, A002859, A049821, A049878, A049879.

# First we modify Peter Luschny's program from A002858 (with len >= 3):
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; end if; if hr <= hu then return evalb(h = 1); end if; if hr + hu = n then tu := behead(u); tr := behead(r); return isUlam(n, h + 1, tu, tr); end if; if hr + hu < n then tu := behead(u); return isUlam(n, h, tu, r); end if; tr := behead(r); isUlam(n, h, u, tr); end proc; nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len - 1 then return [op(u), n]; end if; nextUlam(n + 1, [op(u), n], [n, op(r)]); else nextUlam(n + 1, u, r); end if; end proc; nextUlam(3, [1, 3], [3, 1]); end proc:
# Next we create a function to calculate a(n) for given n >= 3:
a := proc(n) local u, a, i, j: u := 0: if 3 <= n then a := UlamList(n): for i to n - 2 do for j from i + 1 to n - 1 do if a[n] = a[i] + a[j] then u := max(i, j): end if: end do: end do: end if: u: end proc:
# Finally, we create a list of values for a(n):
seq(a(n), n=3..100); # Petros Hadjicostas, Nov 20 2019

Name edited by and more terms from Petros Hadjicostas, Nov 20 2019

A049878 a(n) = min(j,k), where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 3, 5, 4, 3, 5, 4, 3, 4, 3, 5, 4, 3, 4, 3, 5, 4, 3, 15, 1, 2, 8, 3, 11, 3, 14, 3, 2, 3, 5, 19, 21, 3, 24, 3, 17, 4, 3, 4, 3, 27, 3, 27, 3, 4, 27, 3, 27, 3, 4, 3, 5, 27, 3, 4, 27, 3, 27, 3, 27, 3, 27, 3, 36, 2, 3, 5, 27, 14, 27, 3, 27, 3, 27, 3, 4, 3, 27, 3, 4, 27, 3

Offset: 3

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Nov 20 2019: (Start)
A002859(3) = 4 = 1 + 3 = A002859(1) + A002859(2), so a(3) = min(1,2) = 1.
A002859(4) = 5 = 1 + 4 = A002859(1) + A002859(3), so a(4) = min(1,3) = 1.
A002859(5) = 6 = 1 + 5 = A002859(1) + A002859(4), so a(5) = min(1,4) = 1.
A002859(6) = 8 = 3 + 5 = A002859(2) + A002859(4), so a(6) = min(2,4) = 2.
A002859(7) = 10 = 4 + 6 = A002859(3) + A002859(5), so a(7) = min(3,5) = 3.
(End)

Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.

Cf. A002858, A002859, A049821, A049877, A049879.

# First we modify Peter Luschny's program from A002858 (with len >= 3):
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; end if; if hr <= hu then return evalb(h = 1); end if; if hr + hu = n then tu := behead(u); tr := behead(r); return isUlam(n, h + 1, tu, tr); end if; if hr + hu < n then tu := behead(u); return isUlam(n, h, tu, r); end if; tr := behead(r); isUlam(n, h, u, tr); end proc; nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len - 1 then return [op(u), n]; end if; nextUlam(n + 1, [op(u), n], [n, op(r)]); else nextUlam(n + 1, u, r); end if; end proc; nextUlam(3, [1, 3], [3, 1]); end proc:
# Next we create a function to calculate a(n) for given n >= 3:
a := proc(n) local u, a, i, j: u := 0: if 3 <= n then a := UlamList(n): for i to n - 2 do for j from i + 1 to n - 1 do if a[n] = a[i] + a[j] then u := min(i, j): end if: end do: end do: end if: u: end proc:
# Finally, we create a list of values for a(n):
seq(a(n), n=3..100); # Petros Hadjicostas, Nov 20 2019

Name edited and typo in the data corrected by Petros Hadjicostas, Nov 20 2019

More terms from Petros Hadjicostas, Nov 20 2019

A049879 a(n) = |j - k|, where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Original entry on oeis.org

1, 2, 3, 2, 2, 3, 4, 6, 4, 7, 9, 7, 10, 12, 12, 14, 12, 15, 17, 17, 19, 17, 20, 22, 5, 26, 25, 19, 27, 16, 29, 13, 31, 33, 33, 31, 8, 6, 37, 3, 39, 19, 40, 42, 42, 44, 4, 46, 6, 48, 48, 8, 50, 11, 53, 53, 55, 53, 16, 58, 58, 17, 60, 19, 63, 23, 65, 25, 67, 11, 70, 70, 68, 33, 58, 35, 74

Offset: 3

Clark Kimberling

nonn

			From _Petros Hadjicostas_, Nov 20 2019: (Start)
A002859(3) = 4 = 1 + 3 = A002859(1) + A002859(2), so a(3) = |1-2| = 1.
A002859(4) = 5 = 1 + 4 = A002859(1) + A002859(3), so a(4) = |1-3| = 2.
A002859(5) = 6 = 1 + 5 = A002859(1) + A002859(4), so a(5) = |1-4| = 3.
A002859(6) = 8 = 3 + 5 = A002859(2) + A002859(4), so a(6) = |2-4| = 2.
A002859(7) = 10 = 4 + 6 = A002859(3) + A002859(5), so a(7) = |3-5| = 2.
(End)

Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.

Cf. A002858, A002859, A049821, A049877, A049878.

# First we modify Peter Luschny's program from A002858 (with len >= 3):
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; end if; if hr <= hu then return evalb(h = 1); end if; if hr + hu = n then tu := behead(u); tr := behead(r); return isUlam(n, h + 1, tu, tr); end if; if hr + hu < n then tu := behead(u); return isUlam(n, h, tu, r); end if; tr := behead(r); isUlam(n, h, u, tr); end proc; nextUlam := proc(n, u, r) if isUlam(n, 0, u, r) then if nops(u) = len - 1 then return [op(u), n]; end if; nextUlam(n + 1, [op(u), n], [n, op(r)]); else nextUlam(n + 1, u, r); end if; end proc; nextUlam(3, [1, 3], [3, 1]); end proc:
# Next we create a function to calculate a(n) for given n >= 3:
a := proc(n) local u, a, i, j: u := 0: if 3 <= n then a := UlamList(n): for i to n - 2 do for j from i + 1 to n - 1 do if a[n] = a[i] + a[j] then u := abs(i-j): end if: end do: end do: end if: u: end proc:
# Finally, we create a list of values for a(n):
seq(a(n), n=3..100); # Petros Hadjicostas, Nov 20 2019

Name edited by and more terms from Petros Hadjicostas, Nov 20 2019

A081026 Variation on Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = smallest (n odd) or largest (n even) number > a(n-1) that is a unique sum of two distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 12, 23, 24, 47, 48, 95, 96, 191, 192, 383, 384, 767, 768, 1535, 1536, 3071, 3072, 6143, 6144, 12287, 12288, 24575, 24576, 49151, 49152, 98303, 98304, 196607, 196608, 393215, 393216, 786431, 786432, 1572863, 1572864, 3145727

Offset: 1

David W. Wilson, Mar 02 2003

nonn

Dan Asimov, post to math-fun mailing list, Feb 11, 2003.

Cf. A002858, A081025.

Appears that a(2k) = 3*2^(k-1)-1, a(2k+1) = 3*2^(k-1) for k >= 1.

cp's OEIS Frontend

A214603 Location of the first gap of exactly n in Ulam numbers, or zero if none is known. The zero terms are conjectural.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A307331 The number of Ulam numbers below 10^n.

Views

Author

Keywords

Comments

Links

Crossrefs

A378795 Ulam numbers that are semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A006844 a(1)=4, a(2)=5; thereafter a(n) is smallest number that is greater than a(n-1) and having a unique representation as a(j) + a(k) for j

Views

Author

Keywords

Comments

References

Links

Programs

Haskell

Mathematica

Formula

A048951 (2,4) Ulam sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

A049821 a(n) = j + k, where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A049877 a(n) = max(j,k), where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A049878 a(n) = min(j,k), where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Extensions

A049879 a(n) = |j - k|, where u(n) = u(j) + u(k) is the unique sum of Ulam numbers described in A002859 (with 1 <= j < k < n).

Views

Author