A007300 -id:A007300 - cp's OEIS frontend

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

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

A003664 a(n) is smallest number larger than a(n-1) and not = a(j) + a(k), j < k.

Original entry on oeis.org

2, 5, 6, 9, 10, 13, 17, 20, 21, 24, 28, 32, 35, 36, 39, 43, 47, 50, 51, 54, 58, 62, 65, 66, 69, 73, 77, 80, 81, 84, 88, 92, 95, 96, 99, 103, 107, 110, 111, 114, 118, 122, 125, 126, 129, 133, 137, 140, 141, 144, 148, 152, 155, 156, 159, 163, 167, 170, 171, 174, 178, 182

Offset: 1

N. J. A. Sloane, Mira Bernstein

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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 117.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Steven R. Finch, Are 0-Additive Sequences Always Regular?, Am. Math. Monthly 99 (7) (1992) 671-673.
R. K. Guy, s-Additive sequences, Preprint, 1994. (Annotated scanned copy)
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A007300.

I:=[2, 5, 6, 9, 10, 13, 17, 20, 21, 24, 28]; [n le 11 select I[n] else Self(n-1)+Self(n-5)-Self(n-6): n in [1..60]]; // Vincenzo Librandi, Jul 10 2014

Sort[Join[{10},Flatten[Table[15*n+{2,5,6,9,13},{n,0,12}]]]] (* Harvey P. Dale, Jul 12 2012 *)
CoefficientList[Series[x(x^10 + 2*x^9 - 2 x^8 + 2 x^7 + x^6 + x^5 + x^4 + 3 x^3 + x^2 + 3 x + 2)/((x - 1)^2 (x^4 + x^3 + x^2 + x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Jul 10 2014 *)

Vec(x*(x^10+2*x^9-2*x^8+2*x^7+x^6+x^5+x^4+3*x^3+x^2+3*x+2)/((x-1)^2*(x^4+x^3+x^2+x+1)) + O(x^100)) \\ Colin Barker, Jul 09 2014

The numbers 2, 5, 6, 9, 13 mod 15, plus the number 10. - Ralf Stephan, Mar 28 2004

G.f.: x*(x^10 +2*x^9 -2*x^8 +2*x^7 +x^6 +x^5 +x^4 +3*x^3 +x^2 +3*x +2) / ((x -1)^2*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Jul 09 2014

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

A117140 a(1)=5, a(2)=7; for n >= 3, a(n) is smallest number which is uniquely of the form a(j)+a(k) with 1<=j

Original entry on oeis.org

5, 7, 12, 17, 19, 22, 26, 27, 32, 33, 36, 37, 40, 42, 46, 50, 56, 60, 66, 70, 71, 74, 81, 85, 91, 99, 101, 132, 138, 140, 150, 154, 156, 158, 160, 179, 181, 185, 189, 195, 205, 209, 233, 234, 236, 240, 248, 258, 264, 268, 287, 289, 293, 297, 303, 336, 340, 342, 348

Offset: 1

N. J. A. Sloane, Apr 29 2006

nonn

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

Does not seem to have any nice periodicity like A007300 does. - Joshua Zucker, May 24 2006

T. D. Noe, Table of n, a(n) for n=1..10000
A. N. Debono, Ulam Sequences
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

a117140 n = a117140_list !! (n-1)
a117140_list = 5 : 7 : ulam 2 7 a117140_list
-- Function ulam as defined in A002858.
-- Reinhard Zumkeller, Nov 03 2011

max=100; A117140 = {5, 7}; Do[ AppendTo[ A117140, n = Last[A117140]; While[n++; Length[ DeleteCases[ Intersection[A117140, n - A117140], n/2, 1, 1]] != 2]; n], {max}]; A117140 (* Jean-François Alcover, Nov 30 2011 *)

More terms from Joshua Zucker, May 24 2006

A003669 a(n) is smallest number which is uniquely a(j) + a(k), j < k.

Original entry on oeis.org

3, 4, 7, 10, 11, 13, 15, 16, 21, 22, 27, 30, 35, 36, 41, 44, 50, 53, 55, 61, 69, 70, 75, 78, 84, 87, 92, 93, 98, 101, 107, 112, 118, 121, 132, 135, 138, 141, 149, 150, 164, 166, 169, 175, 178, 189, 192, 195, 197, 198, 206, 212, 215, 223, 232, 252, 254, 255, 263, 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..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

a003669 n = a003669_list !! (n-1)
a003669_list = 3 : 4 : ulam 2 4 a003669_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]]]] &, {3, 4}, 58] (* Michael De Vlieger, Nov 16 2017 *)

A003670 a(n) is smallest number which is uniquely a(j)+a(k).

Original entry on oeis.org

4, 7, 11, 15, 18, 19, 23, 25, 27, 31, 32, 33, 35, 41, 47, 49, 55, 57, 61, 63, 75, 87, 89, 91, 105, 119, 121, 125, 129, 133, 139, 147, 153, 161, 185, 189, 203, 206, 213, 225, 233, 235, 241, 243, 251, 257, 259, 265, 273, 281, 285, 293, 295, 303, 305, 307, 309, 317

Offset: 1

N. J. A. Sloane, Mira Bernstein

nonn

Sequence is periodic mod 11301098.

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.
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
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)
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers

a003670 n = a003670_list !! (n-1)
a003670_list = 4 : 7 : ulam 2 7 a003670_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]]]] &, {4, 7}, 56] (* Michael De Vlieger, Nov 16 2017 *)

A078425 Primes in "Ulam's Prime sequence". A prime is in the sequence iff p+1 can be expressed in exactly 1 way as the sum of 2 previous distinct primes.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 41, 43, 59, 83, 89, 107, 109, 127, 139, 157, 163, 173, 199, 211, 223, 257, 271, 277, 293, 307, 331, 347, 367, 397, 421, 443, 457, 491, 541, 557, 587, 601, 631, 691, 761, 769, 821, 911, 941, 971, 991, 1009, 1033, 1103, 1129, 1153, 1201

Offset: 1

Jon Perry, Dec 29 2002

nonn

a(1) = 3, a(2) = 5; for n >= 3, a(n) is smallest prime which is uniquely a(j) + a(k) - 1, with 1<= j < k < n.
Is the (3,5) sequence finite or infinite? Note that (3,7) as a starting sequence has only 2 terms and (7,11) yields 7, 11, 17, 23, 29 only. Equally using -1 as a rule creates more variants.
The sequence continues at least up to a(2227) = 400031.
After about 500 terms, the graph of this sequences appears almost linear. - T. D. Noe, Jan 20 2008

			a(3)=7 as 8=3+5. a(4)=11 as 12=5+7 (and nothing else).

T. D. Noe, Table of n, a(n) for n=1..10000
Index entries for Ulam numbers

Cf. A002858 (Ulam numbers), A002859, A003666, A003667, A001857, A048951, A007300.

v=vector(1220);vc=2;v[1]=3;v[2]=5; forprime (p=7,1220,p1=p+1;pc=0;fl=0;for (i=1,vc-1, for (j=i+1,vc,if (v[i]+v[j]==p1,pc++);if (pc>1,fl=1);if (fl,break));if (fl,break));if (pc==0,fl=1);if (!fl,vc++;v[vc]=p));print(vecextract(v,concat("1..",vc)))

Edited and extended by Klaus Brockhaus, Apr 14 2005

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

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A003664 a(n) is smallest number larger than a(n-1) and not = a(j) + a(k), j < k.

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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

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A117140 a(1)=5, a(2)=7; for n >= 3, a(n) is smallest number which is uniquely of the form a(j)+a(k) with 1<=j

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A003669 a(n) is smallest number which is uniquely a(j) + a(k), j < k.

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Haskell

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A003670 a(n) is smallest number which is uniquely a(j)+a(k).

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Haskell

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A078425 Primes in "Ulam's Prime sequence". A prime is in the sequence iff p+1 can be expressed in exactly 1 way as the sum of 2 previous distinct primes.

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