A002858 -id:A002858 - cp's OEIS frontend

A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, 565, 593, 662, 775, 822, 916, 970, 1016, 1159, 1312, 1395, 1523, 1572, 1821, 1896, 2029, 2254, 2379, 2510, 2780, 2925, 3155, 3354, 3591, 3797, 3998, 4297, 4433, 4779, 4851

Offset: 1

N. J. A. Sloane and Simon Plouffe

An alternative definition is to start with 1 and then continue with the least number such that all pairwise differences of distinct elements are all distinct. - Jens Voß, Feb 04 2003. [However, compare A003022 and A227590. - N. J. A. Sloane, Apr 08 2016]
Rachel Lewis points out [see link] that S, the sum of the reciprocals of this sequence, satisfies 2.158435 <= S <= 2.158677. Similarly, the sum of the squares of reciprocals of this sequence converges to approximately 1.33853369 and the sum of the cube of reciprocals of this sequence converges to approximately 1.14319352. - Jonathan Vos Post, Nov 21 2004; comment changed by N. J. A. Sloane, Jan 02 2020
Let S denote the reciprocal sum of a(n). Then 2.158452685 <= S <= 2.158532684. - Raffaele Salvia, Jul 19 2014
From Thomas Ordowski, Sep 19 2014: (Start)
Known estimate: n^2/2 + O(n) < a(n) < n^3/6 + O(n^2).
Conjecture: a(n) ~ n^3 / log(n)^2. (End)

			The second term is 2 because the 3 pairwise sums 1+1=2, 1+2=3, 2+2=4 are all distinct.
The third term cannot be 3 because 1+3 = 2+2. But it can be 4, since 1+4=5, 2+4=6, 4+4=8 are distinct and distinct from the earlier sums 1+1=2, 1+2=3, 2+2=4.

P. Erdős and R. Graham, Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique (1980).
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.20.2.
R. K. Guy, Unsolved Problems in Number Theory, E28.
A. M. Mian and S. D. Chowla, On the B_2-sequences of Sidon, Proc. Nat. Acad. Sci. India, A14 (1944), 3-4.
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..5818 (terms less than 2*10^9)
Thomas Bloom, Problem 340, Erdős Problems.
Yin Choi Cheng, Greedy Sidon sets for linear forms, J. Num. Theor. (2024).
Rachel Lewis, Mian-Chowla and B2 sequences, 1999. [Thanks to _Steven Finch_ for providing this document. Included with permission. - _N. J. A. Sloane_, Jan 02 2020]
Kevin O'Bryant, B_h-Sets and Rigidity, arXiv:2312.10910 [math.NT], 2023.
Raffaele Salvia, Table of n, a(n) for n=1...25000
R. Salvia, A New Lower Bound for the Distinct Distance Constant, J. Int. Seq. 18 (2015) # 15.4.8.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
Eric Weisstein's World of Mathematics, B2 Sequence.
Eric Weisstein's World of Mathematics, Chowla Sequence.
Zhang Zhen-Xiang, A B_2-sequence with larger reciprocal sum, Math. Comp. 60 (1993), 835-839.
Index entries for B_2 sequences.

Row 2 of A347570.
Cf. A051788, A080200 (for differences between terms).
Different from A046185. Cf. A011185.
See also A003022, A227590.
A259964 has a greater sum of reciprocals.
Cf. A002858.

import Data.Set (Set, empty, insert, member)
a005282 n = a005282_list !! (n-1)
a005282_list = sMianChowla [] 1 empty where
   sMianChowla :: [Integer] -> Integer -> Set Integer -> [Integer]
   sMianChowla sums z s | s' == empty = sMianChowla sums (z+1) s
                        | otherwise   = z : sMianChowla (z:sums) (z+1) s
      where s' = try (z:sums) s
            try :: [Integer] -> Set Integer -> Set Integer
            try []     s                      = s
            try (x:sums) s | (z+x) `member` s = empty
                           | otherwise        = try sums $ insert (z+x) s
-- Reinhard Zumkeller, Mar 02 2011

a[1]:= 1: P:= {2}: A:= {1}:
for n from 2 to 100 do
  for t from a[n-1]+1 do
    Pt:= map(`+`,A union {t},t);
    if Pt intersect P = {} then break fi
  od:
  a[n]:= t;
  A:= A union {t};
  P:= P union Pt;
od:
seq(a[n],n=1..100); # Robert Israel, Sep 21 2014

t = {1}; sms = {2}; k = 1; Do[k++; While[Intersection[sms, k + t] != {}, k++]; sms = Join[sms, t + k, {2 k}]; AppendTo[t, k], {49}]; t (* T. D. Noe, Mar 02 2011 *)

A005282_vec(N, A=[1], U=[0], D(A, n=#A)=vector(n-1, k, A[n]-A[n-k]))={ while(#A2 && U=U[k-1..-1]);A} \\ M. F. Hasler, Oct 09 2019

aupto(L)= my(S=vector(L), A=[1]); for(i=2, L, for(j=1, #A, if(S[i-A[j]], next(2))); for(j=1, #A, S[i-A[j]]=1); A=concat(A, i)); A \\ Ruud H.G. van Tol, Jun 30 2025

from itertools import count, islice
def A005282_gen(): # generator of terms
    aset1, aset2, alist = set(), set(), []
    for k in count(1):
        bset2 = {k<<1}
        if (k<<1) not in aset2:
            for d in aset1:
                if (m:=d+k) in aset2:
                    break
                bset2.add(m)
            else:
                yield k
                alist.append(k)
                aset1.add(k)
                aset2 |= bset2
A005282_list = list(islice(A005282_gen(),30)) # Chai Wah Wu, Sep 05 2023

a(n) = A025582(n) + 1.

a(n) = (A034757(n)+1)/2.

Examples added by N. J. A. Sloane, Jun 01 2008

A054540 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 19, 31, 34, 53, 118, 171, 289, 323, 441, 612, 730, 1171, 1783, 2513, 4296, 12276, 16572, 20868, 25164, 46032, 48545, 52841, 73709, 78005, 151714, 229719, 537443, 714321, 792326, 944040, 1022045, 1251764, 3755292, 3985011

Offset: 0

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 09 2000; Dec 17 2000

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 3985011. There seems to be a hidden aspect or mystery here: what is it about the more and more harmonious equal temperaments that causes them to express themselves collectively as a perfect, self-accumulating recurrent sequence?
From Eliora Ben-Gurion, Dec 15 2022: (Start)
The answer is because temperament mappings can be added. If harmonic correspondences are written in a bra, that is Example: a tuning with 118 equal steps to the octave has a second harmonic on the 118th step by definition, the third harmonic is approximated with 187 steps, and the fifth is with 274 steps, which leads to <118 187 274]. A 171 equal division system will have a corresponding bra <171 271 397]. When these two are added, we obtain <289 458 671], which is exactly how the 2nd, 3rd, and 5th harmonics are represented in 289 equal divisions of the octave. (End)

			34 = 31 + the earlier term 3. Again, 118 = 53 + the earlier terms 34 and 31.

Tonalsoft - Encyclopedia of Microtonal Music Theory, Val - a linear functional on the vectors in the tonespace lattice.
Xenharmonic Wiki, Val

Cf. A001149, A018065, A001856, A002858, A007335, A060525, A060526, A060527.

Stochastic recurrence rule - the next term equals the current term plus one or more previous terms: a(n+1) = a(n) + a(n-x) + ... + a(n-y) + ... + a(n-z), etc.

A007300 a(1)=2, a(2)=5; for n >= 3, a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

Original entry on oeis.org

2, 5, 7, 9, 11, 12, 13, 15, 19, 23, 27, 29, 35, 37, 41, 43, 45, 49, 51, 55, 61, 67, 69, 71, 79, 83, 85, 87, 89, 95, 99, 107, 109, 119, 131, 133, 135, 137, 139, 141, 145, 149, 153, 155, 161, 163, 167, 169, 171, 175, 177, 181, 187, 193, 195, 197, 205, 209, 211, 213, 215

Offset: 1

N. J. A. Sloane, Mira Bernstein

An Ulam-type sequence - see A002858 for many further references, comments, etc.
I have a note saying that this is periodic mod 126. Is that correct? - N. J. A. Sloane, Apr 29 2006
Comments from Joshua Zucker, May 24 2006: "Concerning the conjecture about periodicity mod 126. Out of the first 300 terms, only the 2 and 12 are even. But if you neglect those first 6 terms, mod 2 they're all odd, mod 9 it goes: 0 4 6 1 7 4 6 8 7 2 4 6 8 5 0 8 1 2 5 7 0 2 4 6 1 5 0 2 8 1 5 7 which appears to repeat indefinitely and mod 7 it goes: 0 2 6 1 3 0 2 6 5 4 6 1 2 6 1 3 5 4 1 2 4 0 5 0 2 4 6 1 5 2 6 1 which also appears to repeat indefinitely.
"So it seems as though neglecting the first few terms, it is indeed periodic mod 126 with period 32. In fact it appears that after the first few terms, a(n+32) = a(n) + 126. But this is only based on the first few hundred terms and is not proved!
"The Mathworld link cites a proof that sequences of this type (2,n) have only two even terms and another proof that sequences with only finitely many even terms must eventually have periodic first differences. So I think the period 32 difference of 126 conjecture may be proved in those references."
Given that the sequence of first differences is periodic with period 32 after the first 6 terms (3,2,2,2,1,1), the repeating digits being p=(2,4,4,4,2,6,2,4,2,2,4,2,4,6,6,2,2,8,4,2,2,2,6,4,8,2,10,12,2,2,2,2), one can calculate the n-th term (n>6) as a(n)=13+floor((n-7)/32)*S(32)+S(n-7 mod 32) where S(k)=sum(p(i),i=1..k): (S(k);k=0..32)=(0, 2, 6, 10, 14, 16, 22, 24, 28, 30, 32, 36, 38, 42, 48, 54, 56, 58, 66, 70, 72, 74, 76, 82, 86, 94, 96, 106, 118, 120, 122, 124, 126). - M. F. Hasler, Nov 25 2007

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
J. Cassaigne and S. R. Finch, A class of 1-additive sequences and additive recurrences, Experimental Mathematics, Volume 4 (1995).
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.
S. R. Finch, On the regularity of certain 1-additive sequences, J. Combin. Theory, A60 (1992), 123-130.
Steven R. Finch, Are 0-Additive Sequences Always Regular?, Am. Math. Monthly 99 (7) (1992) 671-673.
S. Finch, Letter to N. J. A. Sloane, Apr. 1994
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Project Euler, Problem 167: Investigating Ulam Sequences.
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
Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A100729, A003666, A003667, A006844.

Cf. A003664.

a007300 n = a007300_list !! (n-1)
a007300_list = 2 : 5 : ulam 2 5 a007300_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

A007300:=n->if n<7 then [2, 5, 7, 9, 11, 12][n] else floor((n-7)/32)*126+[13, 15, 19, 23, 27, 29, 35, 37, 41, 43, 45, 49, 51, 55, 61, 67, 69, 71, 79, 83, 85, 87, 89, 95, 99, 107, 109, 119, 131, 133, 135, 137][modp(n-7,32)+1] fi; # M. F. Hasler, Nov 25 2007

theList = {2,5}; Print[2]; Print[5]; For[i=1,i <= 500,i++, count=0; For[j=1,j <= Length[theList]-1,j++, For[k=j+1,k <= Length[theList],k++, If[theList[[j]]+theList[[k]] == i,count++ ]; ]; ]; If[count == 1, Print[i]; theList = Append[theList,i]; ]; ]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 08 2006 *)
Nest[Append[#, SelectFirst[Union@ Select[Tally@ Map[Total, Select[Permutations[#, {2}], #1 < #2 & @@ # &]], Last@ # == 1 &][[All, 1]], Function[k, FreeQ[#, k]]]] &, {2, 5}, 59] (* Michael De Vlieger, Nov 16 2017 *)

For n > 6, a(n+32) = a(n) + 126. - T. D. Noe, Jan 21 2008

More terms from Joshua Zucker, May 24 2006

More terms from Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 08 2006

A002859 a(1) = 1, 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

1, 3, 4, 5, 6, 8, 10, 12, 17, 21, 23, 28, 32, 34, 39, 43, 48, 52, 54, 59, 63, 68, 72, 74, 79, 83, 98, 99, 101, 110, 114, 121, 125, 132, 136, 139, 143, 145, 152, 161, 165, 172, 176, 187, 192, 196, 201, 205, 212, 216, 223, 227, 232, 234, 236, 243, 247, 252, 256, 258

Offset: 1

N. J. A. Sloane, Mira Bernstein

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

			7 is missing since 7 = 1 + 6 = 3 + 4; but 8 is present since 8 = 3 + 5 has a unique representation.

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 145-151.
R. K. Guy, Unsolved Problems in Number Theory, Section C4.
R. K. Guy, "s-Additive sequences," preprint, 1994.
C. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 358.
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]
Raymond Queneau, Sur les suites s-additives, J. Combin. Theory A 12(1) (1972), 31-71.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
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]
Eric Weisstein's World of Mathematics, Ulam Sequence.
Wikipedia, Ulam number.
Index entries for Ulam numbers.

Cf. A002858 (version beginning 1,2), A199118, A199119.

a002859 n = a002859_list !! (n-1)
a002859_list = 1 : 3 : ulam 2 3 a002859_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

s = {1, 3}; Do[ AppendTo[s, n = Last[s]; While[n++; Length[ DeleteCases[ Intersection[s, n-s], n/2, 1, 1]] != 2]; n], {60}]; s (* Jean-François Alcover, Oct 20 2011 *)

A007086 Next term is uniquely the sum of 3 earlier terms.

Original entry on oeis.org

1, 2, 3, 6, 9, 10, 11, 12, 28, 29, 30, 53, 56, 57, 80, 82, 104, 105, 107, 129, 130, 132, 154, 155, 157, 179, 180, 182, 204, 205, 207, 229, 230, 232, 254, 255, 257, 279, 280, 282, 304, 305, 307, 329, 330, 332, 354, 355, 357, 379, 380, 382, 404, 405, 407, 429

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

nonn

a(1)=1, a(2)=2, a(3)=3, for n>3, a(n) = least number which is a unique sum of three distinct earlier terms. Written this way, we see that this is to 3 as Ulam number A002858 is to 2. - Jonathan Vos Post

			13 through 27 are not in the sequence because of nonuniqueness: 1+3+9=1+2+10=13, 1+3+10=2+3+9=14, 1+2+12=2+3+10=15, 1+6+9=2+3+11=16, 1+7+9=2+6+9=17, 3+6+9=1+6+11=18, 1+6+12=2+6+11=19, 1+9+10=2+6+12=20, 1+9+11=2+9+10=21, 1+10+11=2+9+11=22, 2+9+12=3+9+11=23, 1+11+12=3+9+12=24, 3+10+12=6+9+10=25, 3+11+12=6+9+11=26, 6+9+12=6+10+11=27. - _Jonathan Vos Post_

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
R. G. Wilson, V, Letter to N. J. A. Sloane, Sep. 1992

Cf. A002858.

Clear[a]; a[n_ /; n <= 3] := n; a[n_] := a[n] = (t = Table[a[i]+a[j]+a[k], {i, 1, n-3}, {j, i+1, n-2}, {k, j+1, n-1}] // Flatten; Complement[Select[t // Tally, #[[2]] == 1&][[All, 1]], Array[a, n-1]] // Sort // First); Array[a, 56] (* Jean-François Alcover, Mar 11 2014 *)

G.f.: (22*x^18 -21*x^17 +x^16 -2*x^13 -7*x^12 -15*x^9 +2*x^8 +2*x^7 -2*x^5 -2*x^4 -x^3 -x^2 -x) / (-x^4+x^3+x-1). Conjectured and verified for n<=1100 - Alois P. Heinz, Jan 04 2011

A183534 Square array of generalized Ulam numbers U(n,k), n>=1, k>=2, read by antidiagonals: U(n,k) = n if n<=k; for n>k, U(n,k) = least number > U(n-1,k) which is a unique sum of k distinct terms U(i,k) with i

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 6, 6, 1, 2, 3, 4, 9, 8, 1, 2, 3, 4, 10, 10, 11, 1, 2, 3, 4, 5, 16, 11, 13, 1, 2, 3, 4, 5, 15, 17, 12, 16, 1, 2, 3, 4, 5, 6, 25, 18, 28, 18, 1, 2, 3, 4, 5, 6, 21, 26, 19, 29, 26, 1, 2, 3, 4, 5, 6, 7, 36, 27, 22, 30, 28, 1, 2, 3, 4, 5, 6, 7, 28, 37, 28, 64, 53, 36

Offset: 1

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

The columns are Ulam-type sequences - see A002858 for further information.  Some of these sequences - but not all - seem to have quite simple generating functions.
U(k+1,k)   = k*(k+1)/2.
U(k+2+j,k) = k^2+j for k>=3 and 0<=jU(2*k+2,k) = k*(3*k-1)/2 for k>=3.

			Square array U(n,k) begins:
  1,  1,  1,  1,  1,  1,  ...
  2,  2,  2,  2,  2,  2,  ...
  3,  3,  3,  3,  3,  3,  ...
  4,  6,  4,  4,  4,  4,  ...
  6,  9, 10,  5,  5,  5,  ...
  8, 10, 16, 15,  6,  6,  ...

Alois P. Heinz, Antidiagonals n = 1..87
Index entries for Ulam numbers

Columns k=2-10 give: A002858, A007086, A183527, A183528, A183529, A183530, A183531, A183532, A183533.

Cf. A135737.

b:= proc(n,i,k,h) option remember;
      local t;
      if n<0 or h<0 then 0
    elif n=0 then `if`(h=0, 1, 0)
    elif i=0 or h=0 then 0
    elif h=1 then t:= v(n, k);
                  `if`(t>0 and t<=i, 1, 0)
             else t:= b(n -U(i, k), i-1, k, h-1);
                  t+ `if`(t>1, 0, b(n, i-1, k, h))
      fi
    end:
v:= proc() 0 end:
U:= proc(n,k) option remember;
      local m;
      if n<=k then v(n,k):= n
              else for m from U(n-1, k)+1
                     while b(m, n-1, k, k)<>1 do od;
                   v(m,k):= n; m
      fi
    end:
seq(seq(U(n, 2+d-n), n=1..d), d=1..12);

b[n_, i_, k_, h_] := b[n, i, k, h] = Module[{t}, Which[n < 0 || h < 0, 0, n == 0, If[h == 0, 1, 0], i == 0 || h == 0, 0, h == 1, t = v[n, k]; If[t > 0 && t <= i, 1, 0], True, t = b[n-U[i, k], i-1, k, h-1]; t+If[t > 1, 0, b[n, i-1, k, h]] ] ]; v[, ] = 0; U[n_, k_] := U[n, k] = Module[{m}, If[n <= k, v[n, k] = n, For[m = U[n-1, k]+1, b[m, n-1, k, k] != 1, m++]; v[m, k] = n; m] ]; Table[Table[U[n, 2+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *)

Ulam(N,k=2,v=0)={ my( a=vector(k,i,i), c );
for( n=k,N-1, for( t=1+a[n],9e9, c=0;
  forvec(v=vector(k,i,[i,n]),sum(j=1,k,a[v[j]])==t & c++>1 & next(2),2);
  c||next; v&print1(t","); a=concat(a,t); break;
)); a}
/* M. F. Hasler */

A135737 Ulam type (1-additive) sequences u[1]=2, u[2]=2n+1, u[k+1] is least unique sum u[i]+u[j]>u[k], 1<=i

Original entry on oeis.org

2, 3, 2, 5, 5, 2, 7, 7, 7, 2, 8, 9, 9, 9, 2, 9, 11, 11, 11, 11, 2, 13, 12, 13, 13, 13, 13, 2, 14, 13, 15, 15, 15, 15, 15, 2, 18, 15, 16, 17, 17, 17, 17, 17, 2, 19, 19, 17, 19, 19, 19, 19, 19, 19, 2, 24, 23, 19, 20, 21, 21, 21, 21, 21, 21, 2, 25, 27, 21, 21, 23, 23, 23, 23, 23, 23, 23

Offset: 1

M. F. Hasler, Nov 26 2007

Any of the sequences u=U(2,2n+1) has u[1]=2 and u[n+4]=4n+4; in between these there are the odd numbers 2n+1,...,4n-3. For n>1 there are no other even terms and the sequence of first differences becomes periodic for k>=t (transient phase), such that u[k] = u[k-floor((k-t)/p)*p] + floor((k-t)/p)*d, where p is the period (cf. A100729) and d the fundamental difference (cf. A100730). See the cross-references, especially A002858, for more information.

			The sequence contains the terms of the table T[n,k] = U(2,2n+1)[k], read by antidiagonals: a[1]=T[1,1]=2, a[2]=T[1,2]=3, a[3]=T[2,1]=2, a[4]=T[1,3]=5,...
n=1: U(2,3)= 2, 3, 5, 7, 8, 9,13,14...
n=2: U(2,5)= 2, 5, 7, 9,11,12,...
n=3: U(2,7)= 2, 7, 9,11,13,...
n=4: U(2,9)= 2, 9,11,...

M. F. Hasler, Table of n, a(n) for n = 1..1000, Apr 10 2025
Project Euler, Problem 167: Investigating Ulam sequences

Cf. A001857 = U(2, 3) = row 1, A007300 = U(2, 5) = row 2, A003668 = U(2, 7) = row 3; A100729-A100730 (period).
Cf. A002858 = U(1, 2): this would be row 0, with u[1], u[2] exchanged.
See also: A002859 = U(1, 3), A003666 = U(1, 4), A003667 = U(1, 5).

ulam(a,b,Nmax=30,i)=a=[a,b]; b=[a[1]+b]; for( k=3,Nmax, i=1; while(( i<#b && b[i]==b[i+1] && i+=2 ) || ( i>1 && b[i]==b[i-1] && i++),); a=concat(a,b[i]); b=vecsort(concat(vecextract(b,Str("^..",i)),vector(k-1,j,a[k]+a[j]))); i=0; for(j=1,#b-2, if( b[j]==b[j+2], i+=1<A135737(Nmax=100)=local(T=vector(sqrtint(Nmax*2)+1,n, ulam(2,2*n+1, sqrtint(Nmax*2)+2-n)),i,j); vector(Nmax,k,if(j>1,T[i++ ][j-- ],j=i+1;T[i=1][j]))

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This sequence starts at a(0)=1, subsequent terms a(n) for n>0 being obtained by selecting the (greatest Ulam number) <= 1+Sum_{i=0..n-1} a(i). This ensures that the sequence is complete because Sum_{i=0..n-1} a(i) >= a(n)-1, for all n>=0 and a(0)=1, is a necessary and sufficient condition for completeness.

Number of Fibonacci binary words of length n and having no subword 1011. A Fibonacci binary word is a binary word having no 00 subword. Example: a(5) = 9 because of the 13 Fibonacci binary words of length 5 the following do not qualify: 11011, 10110, 10111 and 01011. - Emeric Deutsch, May 13 2007

a(1) = 1; for n > 1, a(n) = least number > a(n-1) which is a unique sum of two earlier terms, not necessarily distinct. - Franklin T. Adams-Watters, Nov 01 2011

cp's OEIS Frontend

A348413 a(0) = A002858(1) = 1, followed by the greatest Ulam numbers A002858 to form a complete sequence (see algorithm below).

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A005282 Mian-Chowla sequence (a B_2 sequence): a(1) = 1; for n>1, a(n) = smallest number > a(n-1) such that the pairwise sums of elements are all distinct.

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A033627 0-additive sequence: not the sum of any previous pair.

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A004280 2 together with the odd numbers (essentially the result of the first stage of the sieve of Eratosthenes).

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A054540 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the six simple ratios of musical harmony: 6/5, 5/4, 4/3, 3/2, 8/5 and 5/3.

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A007300 a(1)=2, a(2)=5; for n >= 3, a(n) is smallest number which is uniquely of the form a(j) + a(k) with 1 <= j < k < n.

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