A002858 -id:A002858 - cp's OEIS frontend

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 149, 161, 173, 185, 213, 233, 261, 297, 333, 385, 429, 481, 533, 545, 573, 601, 629, 673, 717, 761, 837, 905, 989, 1033, 1085, 1145, 1197, 1257, 1309, 1337, 1397, 1457, 1525, 1625, 1669

Offset: 0

David Applegate and N. J. A. Sloane, Jul 30 2009

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and
b) Q's outer squares were not considered (that is, satisfied a)) in any previous generation, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
Originally constructed in an attempt to explain the Holladay-Ulam CA shown in Fig. 2 of the 1962 Ulam article. However, as explained on page 222 of that article, the actual rule for that CA (see A151906, A151907) is different from ours.
A170896 and A267190 are also closely related cellular automata.
A151895 and A267190 first differ at n=17, when A267190 turns (12,2) ON even though its outer square (11,1) was considered (not turned ON) in a previous generation. - David Applegate, Jan 30 2016

D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191.

David Applegate, Table of n, a(n) for n = 0..250
David Applegate, The movie version
David Applegate, Illustration of first 10 generations
David Applegate, Illustration of first 20 generations
David Applegate, Illustration of first 32 generations
David Applegate, Illustration of first 64 generations
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.], which is also available at arXiv:1004.3036v2
R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
N. J. A. Sloane, Illustration of initial terms (concentrating on a 90-degree sector)
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
S. 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]

See A170896, A170897 for the original Schrandt-Ulam version.

Cf. A151896 (the first differences), A139250, A151905, A151906, A151907, A267190, A267191.

We do not know of a recurrence or generating function.

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016

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.

Original entry on oeis.org

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

N. J. A. Sloane

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

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.

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

Cf. A100729.

Cf. A199122, A199123.

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

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 *)

More terms from Jud McCranie

A060469 Smallest positive a(n) such that number of solutions to a(n) = a(j)+a(k) j

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 23, 25, 28, 30, 35, 37, 40, 42, 47, 49, 52, 54, 59, 61, 64, 66, 71, 73, 76, 78, 83, 85, 88, 90, 95, 97, 100, 102, 107, 109, 112, 114, 119, 121, 124, 126, 131, 133, 136, 138, 143, 145, 148, 150, 155, 157, 160, 162, 167, 169, 172, 174

Offset: 1

Henry Bottomley, Mar 15 2001

Numbers {1,4,6,11} mod 12 plus {2,3,8}.

			11 is in the sequence since it is 3+8 but no other sum of two distinct terms.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A002858, A033627, A056594, A060469, A060470, A060471, A060472.

Virtually identical to A003662.

LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 23}, 100] (* Paolo Xausa, Mar 04 2024 *)

Vec(x*(2*x^10+x^8+x^7+2*x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^2*(x+1)*(x^2+1)) + O(x^100)) \\ Colin Barker, Feb 27 2015

a(n) = a(n-1)+a(n-4)-a(n-5) for n>9. - Colin Barker, Feb 27 2015
G.f.: x*(2*x^10+x^8+x^7+2*x^6+x^5+x^4+x^3+x^2+x+1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Feb 27 2015
a(n) = (6*n - 22 - (-1)^n + A056594(n) - A056594(n+1))/2 for n > 6. - Stefano Spezia, Mar 11 2025

A068820 Ulam numbers that are primes.

Original entry on oeis.org

2, 3, 11, 13, 47, 53, 97, 131, 197, 241, 409, 431, 607, 673, 739, 751, 983, 991, 1103, 1433, 1489, 1531, 1553, 1709, 1721, 2371, 2393, 2447, 2633, 2789, 2833, 2897, 3041, 3109, 3217, 3371, 3373, 3527, 3547, 3593, 3671, 3691, 4057, 4153, 4211, 4297, 4363

Offset: 1

Naohiro Nomoto, Mar 29 2002

Charles R Greathouse IV, Table of n, a(n) for n = 1..49221
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).

Cf. A002858.

# Function 'UlamList' is defined in A002858.
UlamPrimes := n -> select(k->isprime(k), UlamList(n)):
UlamPrimes(403); # Peter Luschny, Apr 05 2019

A170896 Number of ON cells after n generations of the Schrandt-Ulam cellular automaton on the square grid that is described in the Comments.

Original entry on oeis.org

0, 1, 5, 9, 13, 25, 29, 41, 53, 65, 85, 97, 117, 145, 157, 169, 181, 201, 229, 249, 285, 321, 365, 409, 445, 497, 549, 577, 605, 633, 669, 713, 757, 825, 893, 969, 1045, 1105, 1173, 1241, 1309, 1377, 1437, 1473, 1541, 1609, 1693, 1793, 1869, 1945, 2037, 2105, 2189, 2281, 2381, 2521, 2621, 2753, 2869, 2969, 3053, 3129, 3237, 3377, 3485, 3585, 3685, 3817, 3909

Offset: 0

N. J. A. Sloane, Jan 09 2010

nonn

The cells are the squares of the standard square grid.
Cells are either OFF or ON, once they are ON they stay ON, and we begin in generation 1 with 1 ON cell.
Each cell has 4 neighbors, those that it shares an edge with. Cells that are ON at generation n all try simultaneously to turn ON all their neighbors that are OFF. They can only do this at this point in time; afterwards they go to sleep (but stay ON).
A square Q is turned ON at generation n+1 if:
a) Q shares an edge with one and only one square P (say) that was turned ON at generation n (in which case the two squares which intersect Q only in a vertex not on that edge are called Q's "outer squares"), and
b) Q's outer squares were not turned ON in any previous generation.
c) In addition, of this set of prospective squares of the (n+1)th generation satisfying the previous condition, we eliminate all squares which are outer squares of other prospective squares.
A151895, A151906, and A267190 are closely related cellular automata.

D. Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191.

David Applegate, Table of n, a(n) for n = 0..260 (corrected by Sean A. Irvine)
David Applegate, The movie version
David Applegate, After 20 generations, illustrating a(20)=285 (with the A170897(20)=36 newly created cells shown in blue)
David Applegate, After 26 generations, illustrating a(26)=549 (with the A170897(26)=52 newly created cells shown in blue)
David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.]
Sean A. Irvine, Java program (github)
R. G. Schrandt and S. M. Ulam, On recursively defined geometric objects and patterns of growth [Link supplied by Laurinda J. Alcorn, Jan 09 2010.]
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021
S. 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]

Cf. A139250, A170897 (first differences), A151895, A151896, A151906, A267190.

We do not know of a recurrence or generating function.

Entry (including definition) revised by David Applegate and N. J. A. Sloane, Jan 21 2016

A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 10, 16, 17, 18, 19, 22, 64, 65, 66, 68, 69, 128, 132, 188, 190, 191, 194, 252, 253, 255, 313, 314, 318, 374, 376, 377, 436, 441, 496, 497, 499, 500, 502, 560, 561, 563, 621, 622, 626, 682, 684, 685, 687, 745, 746, 805, 811

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(5) = 10 = 1 + 2 + 3 + 4 = 4*5/2, because it is the least number >4 with a unique sum of 4 distinct earlier terms.
a(6) = 16 = 1 + 2 + 3 + 10 = 4^2, because it is the least number >10 with a unique sum of 4 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Index entries for Ulam numbers

Column k=4 of A183534.

Cf. A002858, A007086, A183528-A183533, A007300, A135737.

Maple
```
# see A183534 for programs.
```

Conjectured G.f.: (-61*x^124-56*x^118+53*x^117+3*x^116 -x^115+2*x^114-65*x^113-58*x^112+57*x^111 -56*x^110-x^109-57*x^108+54*x^106 +3*x^105+58*x^104-52*x^102+50*x^101-55*x^100 +56*x^95-53*x^94-3*x^93+x^92-2*x^91 +2*x^90+x^87-x^86 +41*x^84-51*x^83-66*x^82-58*x^81 -52*x^79+4*x^78-58*x^77 -x^74-x^73-x^72-54*x^71 -6*x^70-59*x^69-x^68-58*x^67-2*x^66 -x^65-2*x^64-56*x^63-4*x^62-x^61 -58*x^60-2*x^59-59*x^58-x^57-x^56 -2*x^55-x^54-x^53-54*x^52-6*x^51 -59*x^50-x^49-58*x^48-2*x^47-x^46 -2*x^45-56*x^44-4*x^43-x^42-58*x^41 -2*x^40-x^39-58*x^38-2*x^37-x^36 -2*x^35-x^34-55*x^33-5*x^32-59*x^31 -x^30-2*x^29-56*x^28-4*x^27-x^26 -58*x^25-2*x^24-x^23-58*x^22-3*x^21 -x^20-2*x^19-56*x^18-4*x^17-59*x^16 -x^15-2*x^14-x^13-x^12-42*x^11 -3*x^10-x^9-x^8-x^7-6*x^6 -6*x^5-x^4-x^3-x^2-x) / (-x^74+x^73+x-1). (This has been verified for n up to 1000.)

A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 55, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 145, 163, 190, 217, 235, 271, 280, 1740, 1741, 1744, 1745, 1799, 1804, 1805, 1824, 1825, 1831, 1859, 1869, 1913, 1914, 3554, 10521, 10522, 10526, 10527, 10537, 10563, 10564

Offset: 1

Jonathan Vos Post and Alois P. Heinz, Jan 05 2011

nonn

An Ulam-type sequence - see A002858 for further information.

			a(11) = 55 = 1 + ... + 10 = 10*11/2, because it is the least number >10 with a unique sum of 10 distinct earlier terms.
a(12) = 100 = 1 + ... + 9 + 55 = 10^2, because it is the least number >55 with a unique sum of 10 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..120
Index entries for Ulam numbers

Column k=10 of A183534.

Cf. A002858, A007086, A183527-A183532, A135737.

Maple
```
# see A183534 for programs.
```

A003666 a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n and a(1) = 1, a(2) = 4.

Original entry on oeis.org

1, 4, 5, 6, 7, 8, 10, 16, 18, 19, 21, 31, 32, 33, 42, 46, 56, 57, 66, 70, 79, 82, 91, 96, 104, 105, 107, 116, 129, 130, 131, 141, 158, 165, 168, 179, 180, 182, 191, 204, 205, 206, 216, 217, 218, 219, 229, 230, 244, 256, 266, 267, 268, 281, 290, 315, 316, 317, 318, 328

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

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

R. K. Guy, "s-Additive sequences", preprint, 1994.
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..10000
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

Cf. A003662, A199120, A199121.

a003666 n = a003666_list !! (n-1)
a003666_list = 1 : 4 : ulam 2 4 a003666_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Total, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {1, 4}, 58] (* Michael De Vlieger, Nov 16 2017 *)

A167408 Orderly numbers: a number n is orderly if there exists some number k > tau(n) such that the set of the divisors of n is congruent to the set {1,2,...,tau(n)} mod k.

Original entry on oeis.org

1, 2, 5, 7, 8, 9, 11, 12, 13, 17, 19, 20, 23, 27, 29, 31, 37, 38, 41, 43, 47, 52, 53, 57, 58, 59, 61, 67, 68, 71, 72, 73, 76, 79, 83, 87, 89, 97, 101, 103, 107, 109, 113, 117, 118, 124, 127, 131, 133, 137, 139, 149, 151, 157, 158, 162, 163, 164, 167, 173, 177, 178, 179

Offset: 1

Andrew Weimholt, Nov 03 2009

   n: {divisors(n)} == {1,2,...,tau(n)} mod k
  -------------------------------------------
{1} == {1} mod 2
{1,2} == {1,2} mod 3
{1,5} == {1,2} mod 3
{1,7} == {1,2} mod 5
{1,2,8,4} == {1,2,3,4} mod 5
{1,9,3} == {1,2,3} mod 7
{1,11} == {1,2} mod 3 or 9
{1,2,3,4,12,6} == {1,2,3,4,5,6} mod 7
{1,13} == {1,2} mod 11
{1,17} == {1,2} mod 3,5, or 15
{1,19} == 1,2 mod 17
{1,2,10,4,5,20} == {1,2,3,4,5,6} mod 7
{1,23} == {1,2} mod 3,7, or 21
{1,27,3,9} == {1,2,3,4} mod 5
{1,29} == {1,2} mod 3,9, or 27
{1,31} == {1,2} mod 29
{1,37} == 1,2 mod 5,7, or 35
{1,2,38,19} == {1,2,3,4} mod 5
{1,41} == {1,2} mod 3,13, or 39
{1,43} == {1,2} mod 41
{1,47} == {1,2} mod 3,5,9,15, or 45
{1,2,52,4,26,13} == {1,2,3,4,5,6} mod 7
{1,53} == {1,2} mod 3,17, or 51
{1,57,3,19} == {1,2,3,4} mod 5
{1,2,58,29} == {1,2,3,4} mod 5
{1,59} == {1,2} mod 3,19, or 57
{1,61} == {1,2} mod 59
{1,67} == {1,2} mod 5,13, or 65
{1,2,17,4,68,34} == {1,2,3,4,5,6} mod 7
{1,71} == {1,2} mod 3,23, or 69
{1,2,3,4,18,6,72,8,9,36,24,12} == {1,2,3,4,5,6,7,8,9,10,11,12} mod 13
{1,73} == {1,2} mod 71
{1,2,38,4,19,76} == {1,2,3,4,5,6} mod 7
{1,79} == {1,2} mod 7,11, or 77
{1,83} == {1,2} mod 3,9,27, or 81
{1,87,3,29} == {1,2,3,4} mod 5
{1,89} == {1,2} mod 3,29, or 87
{1,97} == {1,2} mod 5,19, or 95
The primes other than 3 are orderly.
Numbers of the form 4p are orderly when p is an odd prime congruent to 3,5, or 6 mod 7.
For primes, k values can be p-2 or a divisor of p-2 other than 1.
T. D. Noe observed that for composite orderly numbers, n, k seems to be one of the three values: tau(n)+1, tau(n)+3, tau(n)+4.
The composite numbers with k = tau(n)+4 are of the form p^2, where prime p == 3 mod 7.
The orderly numbers with k = tau(n)+3 come in many forms. See A168003. It appears that tau(n)+3 is a prime with primitive root 2 (A001122).
The forms for composite orderly numbers with k = tau(n)+1 are too numerous to list here, but seem to occur for any prime k > 3.
Let p be any prime. Then p^(m-2) is in this sequence if m is a prime with primitive root p. For example, 2^(m-2) is here for every m in A001122; 3^(m-2) is here for every m in A019334; 5^(m-2) is here for every m in A019335. For every prime p, there appear to be an infinite number of prime powers p^(m-2) here. All these numbers are actually very orderly (A167409) because we can choose k = tau(n)+1. - T. D. Noe, Nov 04 2009

			12 is an orderly number because 12's divisors are 1,2,3,4,6,12 and
   1 == 1 (mod 7)
   2 == 2 (mod 7)
   3 == 3 (mod 7)
   4 == 4 (mod 7)
  12 == 5 (mod 7)
   6 == 6 (mod 7)

A. Weimholt, Table of n, a(n) for n = 1..10000
Bill McEachen, A167408/A002858

Cf. A167409 = very orderly numbers (k = tau(n) + 1).
Cf. A167410 = disorderly numbers = numbers not in this sequence.
Cf. A167411 = minimal k values for the orderly numbers.

orderlyQ[n_] := (For[dd = Divisors[n]; tau = Length[dd]; k = 3, k <= Max[tau + 4, Last[dd] - 2], k++, If[ Union[ Mod[dd, k]] == Range[tau], Return[True]]]; False); Select[ Range[180], orderlyQ] (* Jean-François Alcover, Aug 19 2013 *)

Minor editing by N. J. A. Sloane, Nov 06 2009

Information about the tau(n)+3 orderly numbers corrected by T. D. Noe, Nov 16 2009

A033629 Numbers that are not the sum of two distinct Ulam numbers.

Original entry on oeis.org

23, 25, 33, 35, 43, 45, 67, 92, 94, 96, 111, 121, 136, 143, 160, 165, 170, 172, 187, 194, 204, 226, 231, 248, 265, 270, 280, 287, 292, 297, 302, 304, 314, 331, 336, 346, 348, 353, 368, 380, 397, 407, 419, 424, 446, 463, 468, 473, 475, 480, 490, 495, 507

Offset: 1

Jud McCranie

nonn

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

Ruud H.G. van Tol, Table of n, a(n) for n = 1..11944 (upto 2^16)
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).
Ivano Salvo and Agnese Pacifico, Computing Integer Sequences: Filtering vs Generation (Functional Pearl), arXiv:1807.11792 [cs.PL], 2018.

Cf. A002858.

terms = 1000; ulams = {1, 2};
Do[AppendTo[ulams, n = Last[ulams]; While[n++; Length[ DeleteCases[ Intersection[ ulams, n - ulams], n/2, 1, 1]] != 2]; n], {terms}];
uu = Total /@ Subsets[ulams, {2}] // Union;
Complement[Range[Last[uu]], uu] // Take[#, {3, terms+2}]& (* Jean-François Alcover, Dec 02 2018 *)

aupto(N)= my(S=Vec([1, 1], N), U=[]); for(i=1, N, if(1==S[i], for(j=1, #U, my(t=i+U[j]); if(t>N, break); S[t]++); U=concat(U, i))); Vec(select(x->!x, S, 1)) \\ Ruud H.G. van Tol, Jul 05 2025

cp's OEIS Frontend

A151895 Number of ON cells after n generations of the cellular automaton on the square grid that is described in the Comments.

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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.

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A060469 Smallest positive a(n) such that number of solutions to a(n) = a(j)+a(k) j

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Mathematica

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A068820 Ulam numbers that are primes.

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Maple

A170896 Number of ON cells after n generations of the Schrandt-Ulam cellular automaton on the square grid that is described in the Comments.

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A183527 An Ulam-type sequence: a(n) = n if n<=4; for n>4, a(n) = least number > a(n-1) which is a unique sum of 4 distinct earlier terms.

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A183533 An Ulam-type sequence: a(n) = n if n<=10; for n>10, a(n) = least number > a(n-1) which is a unique sum of 10 distinct earlier terms.

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Maple

A003666 a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n and a(1) = 1, a(2) = 4.

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