A100729 -id:A100729 - cp's OEIS frontend

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.

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

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]))

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

A100730 Fundamental difference of Ulam 1-additive sequence starting U(2,2n+1).

Original entry on oeis.org

126, 126, 1778, 6510, 23622, 510, 507842, 1523526, 8388606, 4194302, 597870, 35791394, 21691754, 2046, 511305630, 45678505642, 51539607546, 640638112422, 2748779069430, 25563645345606, 46912496118442, 80418967640942

Offset: 2

Ralf Stephan, Dec 03 2004

nonn

Max Alekseyev, Table of n, a(n) for n = 2..610
M. Akeran, On some 1-additive sequences
S. R. Finch, Patterns in 1-additive sequences, Experimental Mathematics 1 (1992), 57-63.

Cf. A100729 for the period, A001857 for U(2, 3), A007300 for U(2, 5), A003668 for U(2, 7).

a(n) = 2 * A046932(2*n+2)

2 more terms from Balakrishnan V (balaji.iitm1(AT)gmail.com), Nov 15 2007
Further new terms and b-file from Max Alekseyev, Dec 01 2007
b-file extended by Max Alekseyev, Aug 17 2015

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

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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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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

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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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A100730 Fundamental difference of Ulam 1-additive sequence starting U(2,2n+1).

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

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