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Search: a199118 -id:a199118
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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.
(Formerly M2303 N0909)
+10
10
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
An Ulam-type sequence - see A002858 for many further references, comments, etc.
REFERENCES
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.
LINKS
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
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.
EXAMPLE
7 is missing since 7 = 1 + 6 = 3 + 4; but 8 is present since 8 = 3 + 5 has a unique representation.
MATHEMATICA
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 *)
PROG
(Haskell)
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
CROSSREFS
Cf. A002858 (version beginning 1,2), A199118, A199119.
KEYWORD
nonn,nice
AUTHOR
N. J. A. Sloane, Mira Bernstein
STATUS
approved
A199119
Number of partitions of n into distinct terms of (1,3)-Ulam sequence, cf. A002859.
+10
6
1, 1, 0, 1, 2, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6, 7, 7, 8, 10, 9, 9, 12, 13, 13, 13, 14, 17, 18, 18, 19, 21, 23, 25, 26, 27, 30, 33, 33, 36, 40, 42, 43, 45, 51, 55, 55, 57, 62, 67, 71, 72, 76, 82, 87, 91, 95, 100, 107, 112, 116, 124, 132, 137, 143, 151, 159, 170
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..66.
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers
EXAMPLE
The first terms of A002859 are 1, 3, 4, 5, 6, 8, 10, 12, 17, 21, ...
a(10) = #{10, 6+4, 6+3+1, 5+4+1} = 4;
a(11) = #{10+1, 8+3, 6+5, 6+4+1} = 4;
a(12) = #{12, 8+4, 8+3+1, 6+5+1, 5+4+3} = 5.
PROG
(Haskell)
a199119 = p a002859_list where
p _ 0 = 1
p (u:us) m | m < u = 0
| otherwise = p us (m - u) + p us m
CROSSREFS
A000586; A199118, A199017, A199121, A199123.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 03 2011
STATUS
approved
A199120
Number of partitions of n into terms of (1,4)-Ulam sequence, cf. A003666.
+10
6
1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 13, 17, 20, 25, 30, 38, 44, 54, 63, 77, 90, 107, 124, 148, 171, 202, 231, 271, 310, 360, 412, 477, 542, 622, 705, 809, 915, 1042, 1175, 1335, 1501, 1699, 1905, 2148, 2403, 2702, 3018, 3383, 3768, 4212, 4682, 5223, 5794, 6445
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Table of n, a(n) for n=0..54.
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers
EXAMPLE
The first terms of A003666 are 1, 4, 5, 6, 7, 8, 10, 16, 18, 19, ...
a(7) = #{7, 6+1, 5+1+1, 4+1+1+1, 7x1} = 5;
a(8) = #{8, 7+1, 6+1+1, 5+1+1+1, 4+4, 4+1+1+1+1, 8x1} = 7;
a(9) = #{8+1, 7+1+1, 6+1+1+1, 5+4, 5+1+1+1+1, 4+4+1, 4+5x1, 9x1} = 8.
PROG
(Haskell)
a199120 = p a003666_list where
p _ 0 = 1
p us'@(u:us) m | m < u = 0
| otherwise = p us' (m - u) + p us m
CROSSREFS
Cf. A000607; A199016, A199118, A199121, A199122.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 03 2011
STATUS
approved
A199122
Number of partitions of n into terms of (2,3)-Ulam sequence, cf. A001857.
+10
6
1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 11, 14, 16, 20, 23, 29, 33, 39, 47, 54, 64, 75, 86, 101, 117, 135, 155, 179, 204, 236, 268, 306, 349, 397, 450, 511, 577, 653, 736, 831, 934, 1050, 1179, 1322, 1478, 1657, 1848, 2065, 2302, 2562, 2852, 3172, 3518, 3909
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Table of n, a(n) for n=0..56.
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers
EXAMPLE
The first terms of A001857 are 2, 3, 5, 7, 8, 9, 13, 14, 18, 19, ...
a(10) = #{8+2, 7+3, 5+5, 5+3+2, 3+3+2+2, 2+2+2+2+2} = 6;
a(11) = #{9+2, 8+3, 7+2+2, 5+3+3, 5+2+2+2, 3+3+3+2, 3+2+2+2+2} = 7;
a(12) = #{9+3, 8+2+2, 7+5, 7+3+2, 5+5+2, 5+3+2+2, 3+3+3+3, 3+3+2+2+2, 6x2} = 9.
MATHEMATICA
nmax = 60;
U = {2, 3};
Do[AppendTo[U, k = Last[U]; While[k++; Length[DeleteCases[Intersection[U, k - U], k/2, 1, 1]] != 2]; k], {nmax}];
a[n_] := IntegerPartitions[n, All, Select[U, # <= n &]] // Length;
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Oct 12 2021 *)
PROG
(Haskell)
a199122 = p a001857_list where
p _ 0 = 1
p us'@(u:us) m | m < u = 0
| otherwise = p us' (m - u) + p us m
CROSSREFS
Cf. A000607; A199123, A199016, A199118, A199120.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 03 2011
STATUS
approved
A199016
Number of partitions of n into terms of (1,2)-Ulam sequence, cf. A002858.
+10
5
1, 1, 2, 3, 5, 6, 10, 12, 18, 22, 30, 37, 50, 60, 78, 94, 120, 143, 179, 213, 262, 309, 376, 440, 531, 618, 737, 855, 1012, 1167, 1372, 1575, 1840, 2104, 2442, 2783, 3214, 3649, 4193, 4746, 5430, 6126, 6980, 7853, 8914, 10002, 11311, 12660, 14274, 15934
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
LINKS
Table of n, a(n) for n=0..49.
Eric Weisstein's World of Mathematics, Ulam Sequence
Wikipedia, Ulam number
Index entries for Ulam numbers
EXAMPLE
The first terms of A002858 are 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, ...
a(6) = #{6, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, 1+1+1+1+1+1} = 10;
a(7) = #{6+1, 4+3, 4+2+1, 4+1+1+1, 3+3+1, 3+2+2, 3+2+1+1, 3+1+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 12.
PROG
(Haskell)
199016 = p a002858_list where
p _ 0 = 1
p us'@(u:us) m | m < u = 0
| otherwise = p us' (m - u) + p us m
CROSSREFS
Cf. A000607; A199017, A199118, A199120, A199122.
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Nov 03 2011
STATUS
approved
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