A001149 -id:A001149 - cp's OEIS frontend

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.

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 *)

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

A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

Original entry on oeis.org

1, 2, 3, 4, 12, 14, 15, 18, 19, 23, 27, 45, 68, 72, 99, 171, 346, 445, 517, 616, 688, 787, 1133, 1304, 3912, 7136, 8440, 9744, 11048, 12352, 18355, 19659, 20963, 22267, 26795, 28099, 29403, 30707, 40451, 41755, 69854, 71158, 72462, 143620, 216082

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 12 2001

nonn

The sequence was found by a computer search of all of the equal divisions of the octave from 1 to over 216082. The self-accumulating nature of this sequence fails once, between the fourth and fifth terms. The sequence therefore does not meet the rigorous definition of 'impeccable' recurrence. The otherwise perfect recurrence in this sequence is of the type seen in sequences A054540, A060526 and A060527. The numerical value of each term represents a musical scale based on an equal division of the octave. 12, for example, signifies the scale which is formed by dividing the octave into 12 equal parts.

A054540, A060525, A060526, A060527, A060528, A001149, A000045.

A001463 Partial sums of A001462; also a(n) is the last occurrence of n in A001462.

Original entry on oeis.org

1, 3, 5, 8, 11, 15, 19, 23, 28, 33, 38, 44, 50, 56, 62, 69, 76, 83, 90, 98, 106, 114, 122, 131, 140, 149, 158, 167, 177, 187, 197, 207, 217, 228, 239, 250, 261, 272, 284, 296, 308, 320, 332, 344, 357, 370, 383, 396, 409, 422, 436, 450, 464, 478, 492, 506, 521, 536, 551, 566, 581, 596

Offset: 1

N. J. A. Sloane

Vardi gives several identities satisfied by A001463 and this sequence.

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).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Marcus and N. J. Fine, Solutions to Problem 5407, Amer. Math. Monthly 74 (1967), 740-743.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
J. L. Rémy, Sur la suite autodécrite de Golomb, J. Number Theory, vol. 66 1997 pp. 1-28.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
I. Vardi, The error term in Golomb's sequence, J. Number Theory, 40 (1992), 1-11. (See also the Math. Review, 93d:11103)

a001463 n = a001463_list !! (n-1)
a001463_list = scanl1 (+) a001462_list
-- Reinhard Zumkeller, Apr 28 2012

Accumulate[a[1]=1;a[n_]:=a[n]=1+a[n-a[a[n-1]]];Table[a[n],{n,84}]] (* Harvey P. Dale, Oct 20 2011, from Robert G. Wilson v's program in A001463 *)

a(n) is asymptotic to tau^(1-tau)*n^tau where tau is the golden ratio, tau=(1+sqrt(5))/2. More precisely, a(n)= tau^(1-tau)*n^tau + c*n^(1/tau)+0(n^(1/tau)) where c is a constant around 0.6. Is there any known value for c? - Benoit Cloitre, Oct 29 2002

A060233 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to six complementary pairs of ratios which generate simple musical tones (scale steps): 8/7 and 7/4, 6/5 and 5/3, 16/13 and 13/8, 5/4 and 8/5, 4/3 and 3/2 and 11/8 and 16/11.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 10, 15, 19, 22, 24, 26, 31, 37, 41, 46, 50, 53, 72, 84, 87, 130, 137, 140, 171, 183, 217, 224, 270, 494, 764, 851, 1038, 1282, 1308, 1578, 2190, 2684, 3395, 4843, 5004, 5585, 6079, 8269, 14124, 14618, 17302, 20203, 22887, 31737

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), Apr 14 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to over 31737. The numerical value of each term represents a musical scale based on an equal division of the octave. 19, for example, signifies the scale which is formed by dividing the octave into 19 equal parts.

			6 = 4 + the previous term 2. Again, 48545 = 46625 + the previous terms (1578 + 270 + 72).

A054540, A060525, A060526, A060527, A060528, A060529, A001149, A000045.

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

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Original entry on oeis.org

1, 2, 4, 8, 16, 21, 42, 51, 102, 112, 224, 235, 470, 486, 972, 990, 1980, 2002, 4004, 4027, 8054, 8078, 16156, 16181, 32362, 32389, 64778, 64806, 129612, 129641, 259282, 259313, 518626, 518658, 1037316, 1037349, 2074698, 2074734, 4149468

Offset: 1

N. J. A. Sloane

This is a B_2 sequence. More economical recursion: a(1)=1, a(2n)=2a(2n-1), a(2n+1)=a(2n)+r(n), where r(n) is the smallest positive integer not of the form a(j)-a(i) with 1<=iA247556. - Thomas Ordowski, Sep 28 2014

R. K. Guy, Unsolved Problems in Number Theory, E25.
W. Sierpiński, Elementary Theory of Numbers. Państ. Wydaw. Nauk., Warsaw, 1964, p. 444.
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).

T. D. Noe, Table of n, a(n) for n = 1..1000
R. L. Graham, Problem E1910, Amer. Math. Monthly, 73 (1966), 775.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
R. K. Guy and N. J. A. Sloane, Correspondence, 1988.
M. Hall, Cyclic projective planes, Duke Math. J., 4 (1947), 1079-1090.
C. B. A. Peck, Remark on Problem E1910, Amer. Math. Monthly, 75 (1968), 80-81.
W. Sierpiński, Elementary Theory of Numbers, Warszawa 1964.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)

Cf. A054540, A004978, A247556.

a[1] = 1; a[2] = 2; a[n_?OddQ] := a[n] = 2*a[n-1]; a[n_?EvenQ] := a[n] = a[n-1] + r[(n-2)/2]; r[n_] := ( diff = Table[a[j] - a[i], {i, 1, 2*n+1}, {j, i+1, 2*n+1}] // Flatten // Union; max = diff // Last; notDiff = Complement[Range[max], diff]; If[notDiff == {}, max+1, notDiff // First]); Table[a[n], {n, 1, 39}] (* Jean-François Alcover, Dec 31 2012 *)

a(1)=1, a(2)=2, a(2n+1) = 2a(2n), a(2n+2) = a(2n+1) + r(n), where r(n) = smallest positive number not of form a(j) - a(i) with 1 <= i < j <= 2n+1.

More terms from Larry Reeves (larryr(AT)acm.org), Sep 14 2000

A061416 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 11/8 and 16/11 which generate two complementary musical tones.

Original entry on oeis.org

1, 2, 6, 7, 9, 11, 13, 24, 37, 505, 542, 579, 616, 653, 690, 727, 764, 801, 838, 875, 912, 949, 986, 1935, 2921, 4856, 11647, 16503, 148527, 181533, 214539, 219395, 235898, 252401, 268904, 285407, 301910, 318413, 334916, 351419, 367922, 384425

Offset: 1

Mark William Rankin (MarkRankin95511(AT)Yahoo.com), May 02 2001

nonn

The sequence was found by a computer search of all the equal divisions of the octave from 1 to 384425. The numerical value of each term represents a musical scale based on an equal division of the octave. 24, for example, signifies the scale of quartertones which is formed by dividing the octave into 24 equal parts. The recurrence in this sequence breaks down three times, between the 2nd and 3rd terms, between the 9th and 10th terms and between the 28th and 29th terms, but the sequence is of interest because shows the terms generated when this pair of target ratios stands alone. Later, in other sequences, this pair of target ratios will appear in combination with other pairs of target ratios, resulting in new, different, composite sequences. The examples of proper recurrence which do occur in this sequence are of the same type as is seen in sequences A054540, A060526, A060527, A060529 and A060233.

A054540, A060526, A060527, A060529, A060233, A001149 and A000045.

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A060529 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of three complementary pairs of simple musical tones: 7/6 and 12/7, 6/5 and 5/3 and 7/5 and 10/7.

Views

Author

Keywords

Comments

Crossrefs

A001463 Partial sums of A001462; also a(n) is the last occurrence of n in A001462.

Views

Author

Keywords

Comments

References

Links

Programs

Haskell

Mathematica

Formula

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A001856 A self-generating sequence: every positive integer occurs as a(i)-a(j) for a unique pair i,j.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A061416 A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the pair of ratios 11/8 and 16/11 which generate two complementary musical tones.

Views

Author

Keywords

Comments

Crossrefs