A002858 -id:A002858 - cp's OEIS frontend

A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

1, 2, 3, 4, 5, 6, 7, 8, 9, 45, 81, 82, 83, 84, 85, 86, 87, 88, 89, 117, 133, 153, 177, 189, 221, 225, 1325, 1326, 1328, 1329, 1373, 1378, 1379, 1391, 1392, 1398, 1423, 1427, 2717, 2718, 4031, 4032, 4035, 4037, 4039, 5316, 5319, 5346, 5352, 5353, 5354, 5361

Offset: 1

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

nonn

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

			a(10) = 45 = 1 + ... + 9 = 9*10/2, because it is the least number >9 with a unique sum of 9 distinct earlier terms.
a(11) = 81 = 1 + ... + 8 + 45 = 9^2, because it is the least number >45 with a unique sum of 9 distinct earlier terms.

Alois P. Heinz, Table of n, a(n) for n = 1..200
Index entries for Ulam numbers

Column k=9 of A183534.

Cf. A002858, A007086, A183527-A183533, A135737.

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

A003667 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) = 5.

Original entry on oeis.org

1, 5, 6, 7, 8, 9, 10, 12, 20, 22, 23, 24, 26, 38, 39, 40, 41, 52, 57, 69, 70, 71, 82, 87, 98, 102, 113, 119, 129, 130, 133, 144, 160, 161, 162, 163, 175, 196, 205, 208, 209, 222, 223, 224, 226, 237, 253, 254, 255, 256, 268, 269, 270, 271, 272, 284, 285, 286, 303, 318

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

An Ulam-type sequence - see A002858 for many further references, comments, etc.

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

a003667 n = a003667_list !! (n-1)
a003667_list = 1 : 5 : ulam 2 5 a003667_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, 5}, 58] (* Michael De Vlieger, Nov 16 2017 *)

Name clarfied by David A. Corneth, Mar 13 2023

A267190 Number of ON cells after n generations of the 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, 149, 161, 173, 193, 221, 241, 277, 313, 357, 401, 437, 489, 541, 553, 581, 609, 645, 689, 733, 801, 869, 945, 1021, 1081, 1149, 1217, 1277, 1345, 1397, 1433, 1501, 1569, 1653, 1753, 1829, 1905, 1997, 2057, 2141, 2225, 2317, 2449, 2549, 2681, 2797, 2889, 2965, 3041, 3149, 3289

Offset: 0

David Applegate and N. J. A. Sloane, Jan 21 2016

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, and
c) Q's outer squares are not prospective squares of the (n+1)st generation satisfying a).
A151895, A151906, and A170896 are closely related cellular automata.
The key difference between this and A170896 is that if we have two squares Q1 and Q2, both satisfying a), and that are each an outer square of the other, where Q1 satisfies b), but Q2 does not, then for A170896 Q1 is accepted, but for this sequence Q1 is eliminated.  This first happens at n=14, when, for example, A170896 turns (8,3) ON but A267190 doesn't (because (9,2) fails to satisfy b) because (8,1) is ON). - David Applegate, Jan 30 2016
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..260
David Applegate, The movie version
David Applegate, After 16 generations, illustrating a(16)=173 (with the A267191(16)=12 newly created cells shown in blue)
David Applegate, After 36 generations, illustrating a(36)=1021 (with the A267191(36)=76 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.], which is also available at arXiv:1004.3036v2
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]
Index entries for sequences related to toothpick sequences
Index entries for sequences related to cellular automata

Cf. A267191 (first differences), A151895, A151906, A170896.

A003668 a(n) is smallest number which is uniquely a(j)+a(k), j

Original entry on oeis.org

2, 7, 9, 11, 13, 15, 16, 17, 19, 21, 25, 29, 33, 37, 39, 45, 47, 53, 61, 69, 71, 73, 75, 85, 89, 101, 103, 117, 133, 135, 137, 139, 141, 143, 145, 147, 151, 155, 159, 163, 165, 171, 173, 179, 187, 195, 197, 199, 201, 211, 215, 227, 229, 243, 259, 261, 263, 265, 267, 269

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..1000
M. Akeran, On some 1-additive sequences
J. Cassaigne and S. R. Finch, A class of 1-additive sequences and additive recurrences
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. A100729.

a003668 n = a003668_list !! (n-1)
a003668_list = 2 : 7 : ulam 2 7 a003668_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]]]] &, {2, 7}, 58] (* Michael De Vlieger, Nov 16 2017 *)

def aupton(terms):
  alst = [2, 7]
  for n in range(2, terms):
    sums = [alst[j]+alst[k] for j in range(n-1) for k in range(j+1, n)]
    alst.append(min([s for s in sums if sums.count(s)==1 and s > alst[-1]]))
  return alst
print(aupton(60)) # Michael S. Branicky, Feb 07 2021

Akeran gives a formula.

For n>7, a(n+26)=a(n)+126. - T. D. Noe, Jan 21 2008

A068791 Ulam numbers such that 2*n is also an Ulam number.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 18, 36, 53, 57, 69, 206, 273, 400, 983, 1308, 2581, 3205, 4118, 13531, 14892, 88552, 139128, 171243, 278063, 332182, 694354, 3416773, 4311381, 4867024

Offset: 1

Naohiro Nomoto, Mar 29 2002

Cf. A002858, A068799.

More terms from Jud McCranie, Feb 12 2003

A080287 Successively larger gaps in Ulam numbers start at this Ulam number.

Original entry on oeis.org

1, 4, 8, 18, 38, 87, 114, 155, 282, 751, 949, 1257, 1553, 1858, 2178, 4800, 5384, 18796, 37562, 64420, 252719, 933709, 289738117, 332250401, 667752899, 699497052, 966290117, 224582902442, 319654121875, 418843012121, 802386465583

Offset: 1

Jud McCranie, Feb 12 2003

The gaps are in A080288.

			87 and 97 are successive Ulam numbers and this is the first gap of 10 or larger, so 87 is in the sequence.

D. E. Knuth, The Art of Computer Programming, vol 4A, section 7.1.3, exercise 141.

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

Cf. A002858, A080288, A214603, A274522.

Added a(23) found by Don Knuth - Jud McCranie, Aug 22 2008
a(24) on Feb 29 2012; a(25) on Jul 20 2012; and a(26) on Jul 24 2012 by Jud McCranie
a(27) found by Philip Gibbs, Sep 02 2015
a(28) found by Philip Gibbs and Jud McCranie, Sep 09 2015
a(29)-a(31) found by Philip Gibbs, Oct 27 2017
Missing term a(23) added by Jud McCranie, Oct 27 2017

A080288 Successively larger gaps between Ulam numbers.

Original entry on oeis.org

1, 2, 3, 8, 9, 10, 12, 20, 27, 30, 34, 39, 41, 42, 69, 78, 100, 122, 137, 262, 315, 587, 609, 631, 758, 890, 1083, 1092, 1114, 1364, 1373

Offset: 1

Jud McCranie, Feb 12 2003

The lower term of the gaps is in A080287.

			87 and 97 are successive Ulam numbers and this is the first gap of 10 or larger, so 10 is in the sequence.

Philip Gibbs and Judson McCranie, The Ulam Numbers up to One Trillion, (2017).
Shyam Sunder Gupta, Ulam Numbers. In: Exploring the Beauty of Fascinating Numbers. Springer Praxis Books(). Springer, Singapore, (2025).

Cf. A002858, A080287, A214603, A274522.

23rd term found by Don Knuth - Jud McCranie, Aug 22 2008
a(24) on Feb 29 2012; a(25) on Jul 20 2012; and a(26) on Jul 24 2012 found by Jud McCranie;
a(27) found by Philip Gibbs, Sep 02 2015
a(28) found by Philip Gibbs and Jud McCranie, Sep 09 2015
a(29)-a(31) found by Philip Gibbs, Oct 20 2017
Missing a(23) inserted by Jud McCranie, Oct 24 2017

A080328 Smaller of the two Ulam numbers that sum to the n-th Ulam number.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 2, 3, 2, 8, 2, 8, 2, 11, 1, 6, 4, 26, 16, 3, 8, 13, 18, 28, 2, 3, 4, 8, 57, 62, 36, 48, 3, 53, 69, 2, 3, 2, 87, 8, 26, 3, 13, 2, 47, 2, 3, 2, 47, 69, 2, 13, 102, 36, 97, 3, 8, 57, 2, 47, 2, 47, 97, 26, 8, 18, 2, 53, 3, 2, 47, 2, 3, 102, 69, 47, 69, 2, 97, 102

Offset: 1

Jud McCranie, Feb 15 2003

nonn

The first two are zero because the Ulam sequence (A002858) is initialized for those terms. The larger term in the sum is in A080329 and the indices are in A080330 and A080331.

			The 11th Ulam number (26) is the sum of the 6th Ulam number (8) and the 10th Ulam number (18), so a(11)=8.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A002858, A080329, A080330, A080331.

A080329 Larger of the two Ulam numbers that sum to the n-th Ulam number.

Original entry on oeis.org

0, 0, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 36, 47, 47, 53, 36, 53, 69, 69, 69, 69, 69, 97, 99, 102, 106, 69, 69, 102, 97, 145, 102, 106, 175, 177, 180, 102, 189, 180, 206, 206, 219, 189, 236, 238, 241, 206, 189, 258, 260, 180, 273, 219, 316, 316, 282, 339

Offset: 1

Jud McCranie, Feb 15 2003

nonn

The first two are zero because the Ulam sequence (A002858) is initialized for those terms. The smaller term in the sum is in A080328 and the indices are in A080330 and A080331.

			The 11th Ulam number (26) is the sum of the 6th Ulam number (8) and the 10th Ulam number (18), so a(11)=18.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A002858, A080328, A080330, A080331.

A122537 a(1) = 1; for n>1, a(n) is smallest number greater than a(n-1), divisible by n and not equal to any a(i)+a(j) with i and j <= n-1.

Original entry on oeis.org

1, 4, 6, 16, 25, 30, 35, 40, 45, 100, 110, 120, 143, 154, 180, 192, 204, 216, 228, 260, 294, 330, 345, 480, 500, 572, 594, 616, 638, 720, 744, 768, 858, 884, 945, 1008, 1036, 1102, 1131, 1160, 1189, 1218, 1247, 1320, 1395, 1426, 1457, 1584, 1617, 1700, 1734

Offset: 1

J. Lowell, Sep 18 2006

nonn

The definition: "a(1) = 1; for n>1, a(n) is smallest number greater than a(n-1) and not equal to any a(i)+a(j) with i and j <= n-1" produces the odd numbers 1, 3, 5, ...

Jonathan Vos Post asks if 1, 2, 4 and 5 are the only values of n for which n^2 divides a(n), Sep 19 2006. J. Lowell, Oct 02 2006 remarks that n = 1, 2, 4, 5 and 10 have this property and conjectures that there are no other values.

			The 5th term cannot be 20 because 20 = 16+4 and 16 and 4 are both in the sequence.

Cf. A122543 (a(n)/n), A002858, A122544, A122545, A122804.

# a[n] = n-th term of sequence, m[n] = a[n]/n = A122543(n) (Maple program from N. J. A. Sloane)
a:=array(0..100000); m:=array(0..100000); hit:=array(0..100000); B:=100000; M:=100;
for n from 1 to B do hit[n]:=0; od:
a[1]:=1; m[1]:=1; a[2]:=4; m[2]:=2; hit[2]:=1; hit[5]:=1; hit[8]:=1;
for n from 3 to M do i:=n*(floor(a[n-1]/n))+n;
while hit[i] = 1 do i:=i+n; od;
a[n]:= i; m[n]:= i/n;
for j from 1 to n do hit[a[j]+i]:=1; od: od:
[seq(a[n],n=1..M)]; [seq(m[n],n=1..M)];

f[s_] := Block[{n, k},n = Length[s] + 1;k = Last[s] + n - Mod[Last[s], n];While[MemberQ[Union[Plus @@@ Tuples[s, 2]], k], k += n];Append[s, k]];Nest[f, {1}, 51] (* Ray Chandler, Sep 29 2006 *)

More terms from N. J. A. Sloane and Chai Tian (Chao.Tian(AT)epfl.ch), Sep 19 2006

cp's OEIS Frontend

A183532 An Ulam-type sequence: a(n) = n if n<=9; for n>9, a(n) = least number > a(n-1) which is a unique sum of 9 distinct earlier terms.

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A003667 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) = 5.

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A267190 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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A003668 a(n) is smallest number which is uniquely a(j)+a(k), j

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A068791 Ulam numbers such that 2*n is also an Ulam number.

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A080287 Successively larger gaps in Ulam numbers start at this Ulam number.

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A080288 Successively larger gaps between Ulam numbers.

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A080328 Smaller of the two Ulam numbers that sum to the n-th Ulam number.

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A080329 Larger of the two Ulam numbers that sum to the n-th Ulam number.

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