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.

A258264 Numbers having only one representation as a sum of the minimal number of triangular numbers, A000217.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 20, 21, 22, 24, 25, 27, 28, 29, 30, 34, 36, 37, 38, 39, 43, 45, 48, 49, 50, 53, 55, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 73, 78, 79, 83, 84, 87, 88, 90, 91, 92, 93, 97, 99, 100, 101, 102, 105, 108, 110
Offset: 1

Views

Author

Martin Renner, May 25 2015

Keywords

Comments

A258257(a(n)) = 1.
Complement of A258265.

Links

Crossrefs

Cf. A258257, A258265.

Programs

  • Mathematica
    t[n_] := n (n + 1)/2; ok[n_] := Block[{k = 1, t, tt = t /@ Range[ Sqrt[2*n]]}, While[{} == (r = IntegerPartitions[n, {k}, tt]), k++]; Length@ r == 1]; Select[Range[100], ok]

A258265 Numbers having more than one representation as a sum of the minimal number of triangular numbers, A000217.

Original entry on oeis.org

16, 17, 19, 23, 26, 31, 32, 33, 35, 40, 41, 42, 44, 46, 47, 51, 52, 54, 56, 59, 62, 63, 68, 71, 72, 74, 75, 76, 77, 80, 81, 82, 85, 86, 89, 94, 95, 96, 98, 103, 104, 106, 107, 109, 111, 113, 116, 117, 118, 121, 122, 123, 124, 125, 126, 128, 129, 131, 133, 134
Offset: 1

Views

Author

Martin Renner, May 25 2015

Keywords

Comments

A258257(a(n)) > 1.
Complement of A258264.

Links

Crossrefs

Cf. A258257, A258264.

Programs

  • Mathematica
    t[n_] := n*(n + 1)/2; ok[n_] := Block[{k = 1, t, tt = t /@ Range[Sqrt[2*n]]}, While[{} == (r = IntegerPartitions[n, {k}, tt]), k++]; 1 < Length@r]; Select[ Range@ 1000, ok] (* Giovanni Resta, Jun 09 2015 *)

A258258 Least number k having exactly n representations as a sum of the minimal number of triangular numbers, A000217.

Original entry on oeis.org

1, 16, 40, 75, 52, 82, 166, 178, 147, 217, 334, 247, 481, 634, 457, 516, 921, 646, 1047, 1132, 822, 787, 2110, 1351, 1537, 1542, 1402, 1192, 1666, 1696, 2137, 1759, 1876, 2271, 1792, 2712, 2587, 3216, 3909, 2782, 3007, 2956, 4242, 3397, 3682, 4039, 3607, 3601
Offset: 1

Views

Author

Martin Renner, May 24 2015

Keywords

Comments

Fermat's polygonal number theorem states that every positive integer is a sum of at most n n-gonal numbers. The triangular case was proved in 1796 by Gauss (Eureka theorem), stating that every positive integer is the sum of at most three triangular numbers. This sequence is based on this representation as a sum of the minimal number of triangular numbers.

Examples

			a(2) = 16 = 1 + 15 = 6 + 10 is the smallest number with two representations.
a(3) = 40 = 1 + 3 + 36 = 6 + 6 + 28 = 10 + 15 + 15 is the smallest number with three representations.
a(4) = 75 = 3 + 6 + 66 = 3 + 36 + 36 = 10 + 10 + 55 = 15 + 15 + 45 is the smallest number with four representations.

Crossrefs

Cf. A141490, A258257.

A330810 a(n) is the largest number that can be expressed as the sum of three triangular numbers in exactly n ways.

Original entry on oeis.org

53, 194, 470, 788, 1730, 2000, 2693, 4310, 6053, 6845, 10688, 11348, 13970, 12923, 20768, 17135, 27830, 26480, 36245, 31688, 37073, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 82850, 123878, 87890, 119810, 111053, 118490, 118880, 119183
Offset: 1

Views

Author

Jon E. Schoenfield, Jan 01 2020

Keywords

Comments

One or more of the three triangular numbers may be zeros. If it were required that the triangular numbers be positive, sequence A330811 would result.

Crossrefs

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330811.

A330811 a(n) is the largest number that can be expressed as the sum of three positive triangular numbers in exactly n ways.

Original entry on oeis.org

29, 119, 335, 713, 1730, 1328, 3413, 3485, 4565, 6053, 6950, 10688, 11348, 13970, 16778, 20768, 18173, 36245, 26480, 27203, 37073, 35033, 39983, 57860, 46940, 49148, 68258, 62810, 66515, 76985, 73868, 123878, 103403, 87890, 119810, 111053, 118490, 118880
Offset: 0

Views

Author

Jon E. Schoenfield, Jan 01 2020

Keywords

Comments

If the triangular numbers were not required to be positive, sequence A330810 would result.

Crossrefs

Cf. A000217, A002097, A060773, A061262, A063993, A064825, A071530, A111638, A258257, A258258, A330810.
Showing 1-5 of 5 results.

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