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 1-3 of 3 results.

A248232 Numbers k such that A248231(k+1) = A248231(k).

Original entry on oeis.org

1, 4, 8, 11, 15, 18, 22, 25, 28, 32, 35, 39, 42, 46, 49, 52, 56, 59, 63, 66, 69, 73, 76, 80, 83, 87, 90, 93, 97, 100, 104, 107, 110, 114, 117, 121, 124, 128, 131, 134, 138, 141, 145, 148, 151, 155, 158, 162, 165, 168, 172, 175, 179, 182, 186, 189, 192, 196
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

Since A248231(k+1) - A248232(k) is in {0,1} for k >= 1, A248232 and A248233 are complementary.

Examples

			The difference sequence of A248231 is (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, ...), so that A248232 = (1, 4, 8, 11, 15, 18, 22, 25, 28,...) and A248233 = (2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 16, 17,...), the complement of A248232.
		

Crossrefs

Programs

  • Mathematica
    z = 400; p[k_] := p[k] = Sum[1/h^5, {h, 1, k}]; N[Table[Zeta[5] - p[n], {n, 1, z/10}]]
    f[n_] := f[n] = Select[Range[z], Zeta[5] - p[#] < 1/n^4 &, 1]
    u = Flatten[Table[f[n], {n, 1, z}]]   (* A248231 *)
    Flatten[Position[Differences[u], 0]]  (* A248232 *)
    Flatten[Position[Differences[u], 1]]  (* A248233 *)
    Table[Floor[1/(Zeta[5] - p[n])], {n, 1, z}]  (* A248234 *)

A248233 Numbers k such that A248231(k+1) = A248231(k) + 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 53, 54, 55, 57, 58, 60, 61, 62, 64, 65, 67, 68, 70, 71, 72, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 88, 89, 91, 92, 94
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

Since A248231(k+1) - A248232(k) is in {0,1} for k >= 1, A248232 and A248233 are complementary.
This appears to be a duplicate of A097432. - R. J. Mathar, Oct 10 2014

Examples

			The difference sequence of A248231 is (0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, ...), so that A248232 = (1, 4, 8, 11, 15, 18, 22, 25, 28,...) and A248233 = (2, 3, 5, 6, 7, 9, 10, 12, 13, 14, 16, 17,...), the complement of A248232.
		

Crossrefs

Programs

  • Mathematica
    z = 400; p[k_] := p[k] = Sum[1/h^5, {h, 1, k}]; N[Table[Zeta[5] - p[n], {n, 1, z/10}]]
    f[n_] := f[n] = Select[Range[z], Zeta[5] - p[#] < 1/n^4 &, 1]
    u = Flatten[Table[f[n], {n, 1, z}]]   (* A248231 *)
    Flatten[Position[Differences[u], 0]]  (* A248232 *)
    Flatten[Position[Differences[u], 1]]  (* A248233 *)
    Table[Floor[1/(Zeta[5] - p[n])], {n, 1, z}]  (* A248234 *)

A248234 a(n) = floor(1/(zeta(5) - Sum_{h=1..n} 1/h^5)).

Original entry on oeis.org

27, 176, 639, 1706, 3759, 7279, 12842, 21119, 32879, 48986, 70399, 98175, 133466, 177519, 231679, 297386, 376175, 469679, 579626, 707839, 856239, 1026842, 1221759, 1443199, 1693466, 1974959, 2290175, 2641706, 3032239, 3464559, 3941546, 4466175, 5041519
Offset: 1

Views

Author

Clark Kimberling, Oct 05 2014

Keywords

Comments

This sequence provides insight into the manner of convergence of Sum_{h=1..n} 1/h^5.

Crossrefs

Programs

  • Mathematica
    z = 400; p[k_] := p[k] = Sum[1/h^5, {h, 1, k}]; N[Table[Zeta[5] - p[n], {n, 1, z/10}]]
    f[n_] := f[n] = Select[Range[z], Zeta[5] - p[#] < 1/n^4 &, 1]
    u = Flatten[Table[f[n], {n, 1, z}]]   (* A248231 *)
    Flatten[Position[Differences[u], 0]]  (* A248232 *)
    Flatten[Position[Differences[u], 1]]  (* A248233 *)
    Table[Floor[1/(Zeta[5] - p[n])], {n, 1, z}]  (* A248234 *)

Formula

Empirically, a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 5*a(n-4) + 6*a(n-5) - 4*a(n-6) + a(n-7), for n >= 10.
Conjecture (for n >= 3): (12*n*(1+n)*(4+3*n+3*n^2) - 8 - cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/9. - Vaclav Kotesovec, Oct 09 2014
Showing 1-3 of 3 results.