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.

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A205705 Numbers k for which 8 divides prime(k)-prime(j) for some j

Original entry on oeis.org

5, 6, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(5)-p(2)=11-3=8=8*1
p(6)-p(3)=13-5=8=8*1
p(8)-p(2)=19-3=16=8*2
p(8)-p(5)=19-11=8=8*1
p(9)-p(4)=23-7=16=8*2
p(10)-p(3)=29-5=24=8*3

Crossrefs

Cf. A205558, A204892, A204890, A205706, A205710.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]     (* A000040 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204890 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 8; t = d[c]             (* A205704 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]         (* A205705 *)
Table[j[n], {n, 1, z2}]         (* A205706 *)
Table[s[k[n]], {n, 1, z2}]      (* A205707 *)
Table[s[j[n]], {n, 1, z2}]      (* A205708 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205709 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205710 *)

A205712 Numbers k for which 9 divides prime(k)-prime(j) for some j

Original entry on oeis.org

5, 9, 10, 10, 11, 12, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(5)-p(1)=11-2=9=9*1
p(9)-p(3)=23-5=18=9*2
p(10)-p(1)=29-2=27=9*3
p(10)-p(5)=29-11=18=9*2
p(11)-p(6)=31-13=18=9*2
p(12)-p(8)=37-19=18=9*2

Crossrefs

Cf. A205558, A204892, A204890, A205713, A205717.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]        (* A000040 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]        (* A204890 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 9; t = d[c]                (* A205711 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]        (* A205712 *)
Table[j[n], {n, 1, z2}]        (* A205713 *)
Table[s[k[n]], {n, 1, z2}]     (* A205714 *)
Table[s[j[n]], {n, 1, z2}]     (* A205715 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205716 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205717 *)

A205845 [s(k)-s(j)]/3, where the pairs (k,j) are given by A205842 and A205843, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 1, 4, 6, 11, 7, 18, 14, 7, 29, 28, 27, 47, 41, 77, 76, 75, 48, 125, 124, 123, 96, 48, 203, 199, 192, 185, 328, 322, 281, 532, 528, 521, 514, 329, 861, 857, 850, 843, 658, 329, 1393, 1392, 1391, 1364, 1316, 1268, 2254, 2248, 2207, 1926, 3648
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.
The first six terms match these differences:
s(4)-s(2) = 5-2 = 3 = 3*1
s(5)-s(2) = 8-2 = 6 = 3*2
s(5)-s(4) = 8-5 = 3 = 3*1
s(6)-s(1) = 13-1 = 12 = 3*4
s(7)-s(3) = 21-3 = 18 = 3*6
s(8)-s(1) = 34-1 = 33 + 3*11
(See the program at A205842.)

Examples

			The first six terms match these differences:
s(4)-s(2) = 5-2 = 3 = 3*1
s(5)-s(2) = 8-2 = 6 = 3*2
s(5)-s(4) = 8-5 = 3 = 3*1
s(6)-s(1) = 13-1 = 12 = 3*4
s(7)-s(3) = 21-3 = 18 = 3*6
s(8)-s(1) = 34-1 = 33 + 3*11

Crossrefs

Cf. A204890, A205842, A205845.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 3; t = d[c]       (* A205841 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]      (* A205842 *)
Table[j[n], {n, 1, z2}]      (* A205843 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)

A205699 The number j such that 7 divides prime(k)-prime(j), where k(n)=A205698(n).

Original entry on oeis.org

2, 3, 1, 2, 7, 1, 9, 6, 10, 3, 8, 5, 2, 7, 11, 3, 8, 15, 5, 16, 10, 14, 2, 7, 11, 17, 1, 9, 12, 6, 13, 3, 8, 15, 18, 6, 13, 23, 2, 7, 11, 17, 21, 3, 8, 15, 18, 24, 1, 9, 12, 22, 5, 16, 19, 10, 14, 20, 10, 14, 20, 30, 3, 8, 15, 18, 24, 27, 5, 16, 19, 29, 6, 13, 23, 25, 1, 9
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205698.)

Crossrefs

Cf. A204890, A205698, A205558.

Programs

  • Mathematica
    (See the program at A205698.)

A205706 The number j such that 8 divides prime(k)-prime(j), where k(n)=A205705(n).

Original entry on oeis.org

2, 3, 2, 5, 4, 3, 6, 4, 9, 3, 6, 10, 7, 2, 5, 8, 4, 9, 11, 3, 6, 10, 12, 2, 5, 8, 14, 3, 6, 10, 12, 16, 2, 5, 8, 14, 17, 4, 9, 11, 15, 7, 13, 4, 9, 11, 15, 20, 2, 5, 8, 14, 17, 19, 7, 13, 21, 7, 13, 21, 24, 3, 6, 10, 12, 16, 18, 4, 9, 11
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205705.)

Crossrefs

Cf. A204890, A205705, A205558.

Programs

  • Mathematica
    (See the program at A205705.)

A205713 The number j such that 9 divides prime(k)-prime(j), where k(n)=A205712(n).

Original entry on oeis.org

1, 3, 1, 5, 6, 8, 3, 9, 4, 1, 5, 10, 7, 3, 9, 13, 4, 14, 6, 11, 7, 16, 8, 12, 4, 14, 18, 1, 5, 10, 15, 7, 16, 20, 4, 14, 18, 22, 1, 5, 10, 15, 23, 6, 11, 19, 7, 16, 20, 24, 8, 12, 21, 3, 9, 13, 17, 8, 12, 21, 29, 3, 9, 13, 17, 30, 1, 5, 10, 15
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205712.)

Crossrefs

Cf. A204890, A205712, A205558.

Programs

  • Mathematica
    (See the program at A205712.)

A205721 The number j such that 10 divides prime(k)-prime(j), where k(n)=A205720(n).

Original entry on oeis.org

2, 4, 2, 6, 8, 5, 4, 7, 5, 11, 2, 6, 9, 4, 7, 12, 2, 6, 9, 14, 8, 10, 5, 11, 13, 4, 7, 12, 15, 5, 11, 13, 18, 2, 6, 9, 14, 16, 8, 10, 17, 2, 6, 9, 14, 16, 21, 8, 10, 17, 22, 4, 7, 12, 15, 19, 5, 11, 13, 18, 20, 2, 6, 9, 14, 16, 21, 23, 4, 7
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205720.)

Crossrefs

Cf. A204890, A205720, A205558.

Programs

  • Mathematica
    (See the program at A205720.)

A205843 The least number j such that 3 divides s(k)-s(j), where k(n)=A205842(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

2, 2, 4, 1, 3, 1, 6, 1, 6, 8, 2, 4, 5, 3, 7, 2, 4, 5, 10, 2, 4, 5, 10, 12, 1, 6, 8, 9, 3, 7, 11, 1, 6, 8, 9, 14, 1, 6, 8, 9, 14, 16, 2, 4, 5, 10, 12, 13, 3, 7, 11, 15, 2, 4, 5, 10, 12, 13, 18, 2
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205842.)

Crossrefs

Cf. A204890, A205842, A205845.

Programs

  • Mathematica
    (See the program at A205842.)

A205848 The least number j such that 4 divides s(k)-s(j), where k(n)=A205847(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 1, 4, 1, 4, 6, 2, 3, 1, 4, 6, 7, 5, 1, 4, 6, 7, 10, 1, 4, 6, 7, 10, 12, 2, 8, 3, 9, 1, 4, 6, 7, 10, 12, 13, 5, 11, 1, 4, 6, 7, 10, 12, 13, 16, 1, 4, 6, 7, 10, 12, 13, 16, 18, 2, 8, 14, 3, 9, 15
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205847.)

Crossrefs

Cf. A204890, A205847, A205850.

Programs

  • Mathematica
    (See the program at A205847.)

A205853 The least number j such that 5 divides s(k)-s(j), where k(n)=A205852(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

3, 3, 5, 1, 4, 8, 8, 10, 3, 5, 6, 2, 4, 9, 2, 13, 2, 13, 15, 8, 10, 11, 1, 7, 4, 9, 14, 1, 7, 18, 1, 7, 18, 20, 2, 13, 15, 16, 3, 5, 6, 12, 4, 9, 14, 19, 3, 5, 6, 12, 23, 3, 5, 6, 12, 23, 25, 1, 7, 18
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205852.)

Crossrefs

Cf. A204890, A205852, A205840.

Programs

  • Mathematica
    (See the program at A205852.)
Previous Showing 21-30 of 70 results. Next

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