cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

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Author

N. J. A. Sloane, Mira Bernstein

Keywords

Comments

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

Examples

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

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

Crossrefs

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

Programs

  • 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
  • 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 *)

A199119 Number of partitions of n into distinct terms of (1,3)-Ulam sequence, cf. A002859.

Original entry on oeis.org

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
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 03 2011

Keywords

Examples

			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.

Links

Crossrefs

A000586; A199118, A199017, A199121, A199123.

Programs

  • Haskell
    a199119 = p a002859_list where
   p _  0 = 1
   p (u:us) m | m < u = 0
              | otherwise = p us (m - u) + p us m

A199120 Number of partitions of n into terms of (1,4)-Ulam sequence, cf. A003666.

Original entry on oeis.org

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
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 03 2011

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000607; A199016, A199118, A199121, A199122.

Programs

  • 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

A199122 Number of partitions of n into terms of (2,3)-Ulam sequence, cf. A001857.

Original entry on oeis.org

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
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 03 2011

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000607; A199123, A199016, A199118, A199120.

Programs

  • 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
  • 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 *)

A199016 Number of partitions of n into terms of (1,2)-Ulam sequence, cf. A002858.

Original entry on oeis.org

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
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 03 2011

Keywords

Examples

			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.

Links

Crossrefs

Cf. A000607; A199017, A199118, A199120, A199122.

Programs

  • 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
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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