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 1-6 of 6 results.

A184219 a(n) = A184218(n)/A130703(n) unless A130703(n) = 0 in which case a(n) = 0.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 5, 4, 1, 5, 4, 5, 7, 1, 5, 9, 5, 1, 9, 10, 1, 9, 7, 8, 11, 1, 14, 11, 1, 7, 13, 10, 1, 13, 10, 11, 15, 1, 11, 20, 7, 5, 18, 14, 5, 17, 22, 14, 19, 11, 14, 19, 1, 1, 27, 16, 13, 27, 16, 26, 23
Offset: 1

Views

Author

Rémi Eismann, Jan 10 2011

Keywords

Comments

a(n) is the "level" of triangular numbers.
The decomposition of triangular numbers into weight * level + gap is A000217(n) = A130703(n) * a(n) + (n + 1) if a(n) > 0.
A184218(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), 0 otherwise.

Examples

			For n = 3 we have A130703(3) = 0, hence a(3) = 0.
For n = 5 we have A184218(5)/A130703(5) = 9 / 9 = 1, hence a(5) = 1.
For n = 24 we have A184218(24)/A130703(24) = 275 / 55 = 5, hence a(24) = 5.

Links

Crossrefs

Cf. A000217, A000027, A130703, A184218, A118534, A117078, A117563, A001223.

A130882 a(n) = smallest k such that A002808(n+1) = A002808(n) + (A002808(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 4, 7, 2, 4, 5, 13, 2, 7, 4, 19, 2, 4, 23, 2, 5, 2, 13, 4, 31, 2, 3, 2, 17, 37, 2, 19, 4, 43, 2, 4, 47, 2, 7, 2, 5, 53, 2, 5, 2, 4, 29, 61, 2, 3, 2, 4, 67, 2, 4, 5, 73, 2, 3, 2, 4, 79, 2, 4, 83, 2, 5, 2, 43, 89, 2, 7, 2, 3, 2, 47, 97
Offset: 1

Views

Author

Rémi Eismann, Aug 21 2007 - Jan 09 2011

Keywords

Comments

a(n) is the "weight" of composite numbers.
The decomposition of composite numbers into weight * level + gap is A002808(n) = a(n) * A179621(n) + A073783(n) if a(n) > 0.

Examples

			For n = 1 we have A002808(n) = 4, A002808(n+1) = 6; there is no k such that 6 - 4 = 2 = (4 mod k), hence a(1) = 0.
For n = 3 we have A002808(n) = 8, A002808(n+1) = 9; 7 is the smallest k such that 9 - 8 = 1 = (8 mod k), hence a(3) = 7.
For n = 24 we have A002808(n) = 36, A002808(n+1) = 38; 17 is the smallest k such that 38 - 36 = 2 = (36 mod k), hence a(24) = 17.

Links

Crossrefs

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703.

A133150 a(n) = smallest k such that A000290(n+1) = A000290(n) + (A000290(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 14, 23, 17, 47, 31, 79, 49, 119, 71, 167, 97, 223, 127, 41, 46, 359, 199, 439, 241, 527, 82, 89, 337, 727, 391, 839, 449, 137, 73, 1087, 577, 1223, 647, 1367, 103, 217, 94, 1679, 881, 1847, 967, 119, 151, 2207, 1151, 2399, 1249, 113, 193, 401, 1457
Offset: 1

Views

Author

Rémi Eismann, Sep 22 2007 - Jan 10 2011

Keywords

Comments

a(n) is the "weight" of squares (A000290).
The decomposition of squares into weight * level + gap is A000217(n) = a(n) * A184221(n) + A005408(n) if a(n) > 0.

Examples

			For n = 1 we have A000290(n) = 1, A000290(n+1) = 4; there is no k such that 4 - 1 = 3 = (1 mod k), hence a(1) = 0.
For n = 5 we have A000290(n) = 25, A000290(n+1) = 36; 14 is the smallest k such that 36 - 25 = 11 = (25 mod k), hence a(5) = 14.
For n = 18 we have A000290(n) = 324, A000290(n+1) = 361; 41 is the smallest k such that 361 - 324 = 37 = (324 mod k), hence a(18) = 41.

Links

Crossrefs

Cf. A020639, A117078, A117563, A001223, A118534, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

A184218 a(n) = largest k such that A000217(n+1) = A000217(n) + (A000217(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, 104, 119, 135, 152, 170, 189, 209, 230, 252, 275, 299, 324, 350, 377, 405, 434, 464, 495, 527, 560, 594, 629, 665, 702, 740, 779, 819, 860, 902, 945, 989, 1034, 1080, 1127, 1175, 1224, 1274, 1325, 1377
Offset: 1

Views

Author

Rémi Eismann, Jan 10 2011

Keywords

Comments

From the definition, a(n) = A000217(n) - (n + 1) if A000217(n) - (n + 1) > (n + 1), or 0 otherwise, where A000217 are the triangular numbers.

Examples

			For n = 3 we have A000217(3) = 6, A000217(4) = 10; there is no k such that 10 - 6 = 4 = (6 mod k), hence a(3) = 0.
For n = 5 we have A000217(5) = 15, A000217(6) = 21; 9 is the largest k such that 21 - 15 = 6 = (15 mod k), hence a(5) = 9; a(5) = A000217(5) - (5 + 1) = 15 - 6 = 9.
For n = 24 we have A000217(24) = 300, A000217(25) = 325; 275 is the largest k such that 325 - 300 = 25 = (300 mod k), hence a(24) = 275; a(24) = A000217(24) - (24 + 1) = 275.

Links

Crossrefs

Cf. essentially the same as A000096, A000217, A000027, A130703, A184219, A118534, A117078, A117563, A001223.

Programs

  • Magma
    [0,0,0,0] cat [(n+1)*(n-2)/2: n in [5..60]]; // Vincenzo Librandi, Jun 22 2016
  • Mathematica
    Join[{0, 0, 0, 0}, LinearRecurrence[{3, -3, 1}, {9, 14, 20}, 100]] (* G. C. Greubel, Jun 22 2016 *)
lim = 10^4; Table[SelectFirst[Reverse@ Range@ lim, Function[k, PolygonalNumber[n + 1] == # + Mod[#, k] &@ PolygonalNumber@ n]], {n, 53}] /. {k_ /; MissingQ@ k -> 0, k_ /; k == lim -> 0} (* Michael De Vlieger, Jun 30 2016, Version 10.4 *)

Formula

a(n) = (n+1)*(n-2)/2 = A000096(n-2) for n >= 5 and a(n) = 0 for n <= 4. - M. F. Hasler, Jan 10 2011
From Chai Wah Wu, Jun 21 2016: (Start)
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.
G.f.: x^5*(5*x^2 - 13*x + 9)/(1 - x)^3. (End)

A133346 a(n) = smallest k such that prime(n+2) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 7, 11, 0, 15, 21, 21, 31, 7, 11, 35, 9, 17, 17, 61, 9, 21, 23, 23, 77, 7, 19, 97, 101, 91, 19, 13, 41, 25, 127, 47, 139, 21, 17, 31, 11, 167, 13, 37, 11, 61, 25, 39, 7, 13, 73, 9, 227, 25, 239, 35, 15, 9, 29, 271, 269, 37, 25, 7, 61, 59, 27, 21, 13, 11, 113, 113
Offset: 1

Views

Author

Rémi Eismann, Oct 20 2007

Keywords

Examples

			For n = 1 we have prime(n) = 2, prime(n+2) = 5; there is no k such that 5 - 2 = 3 = (2 mod k), hence a(1) = 0.
For n = 6 we have prime(n) = 13, prime(n+2) = 19; 7 is the smallest k such that 19 - 13 = 6 = (13 mod k), hence a(6) = 7.
For n = 30 we have prime(n) = 113, prime(n+2) = 131; 19 is the smallest k such that 131 - 113 = 18 = (113 mod k), hence a(30) = 19.

Links

Crossrefs

Cf. A000040, A117078, A117563, A001223, A118534, A020639, A090369, A090368, A130533, A130650, A130703, A130889, A130882.

A133347 a(n) = smallest k such that prime(n+3) = prime(n) + (prime(n) mod k), or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 17, 19, 27, 29, 27, 33, 39, 47, 49, 55, 59, 19, 61, 65, 15, 29, 31, 31, 29, 29, 89, 23, 113, 41, 121, 15, 27, 47, 21, 17, 31, 15, 33, 61, 25, 57, 57, 193, 71, 43, 31, 43, 221, 73, 233, 27, 83, 257, 37, 29, 51, 51, 21, 11, 97, 289, 41, 313, 107, 67
Offset: 1

Views

Author

Rémi Eismann, Oct 20 2007

Keywords

Examples

			For n = 1 we have prime(n) = 2, prime(n+3) = 7; there is no k such that 7 - 2 = 5 = (2 mod k), hence a(1) = 0.
For n = 10 we have prime(n) = 29, prime(n+3) = 41; 17 is the smallest k such that 41 - 29 = 12 = (29 mod k), hence a(10) = 17.
For n = 53 we have prime(n) = 241, prime(n+3) = 263; 73 is the smallest k such that 263 - 241 = 22 = (241 mod k), hence a(53) = 73.

Links

Crossrefs

Cf. A000040, A117078, A117563, A001223, A118534, A020639, A090369, A090368, A130533, A130650, A130703, A130889, A130882.
Showing 1-6 of 6 results.

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