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 1-10 of 58 results. Next

A205720 Numbers k for which 10 divides prime(k)-prime(j) for some j

Original entry on oeis.org

6, 7, 9, 9, 10, 11, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27
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(6)-p(2)=13-3=10=10*1
p(7)-p(4)=17-7=10=10*1
p(9)-p(2)=23-3=20=10*2
p(9)-p(6)=23-13=10=10*1
p(10)-p(8)=29-19=10=10*1
p(11)-p(5)=31-11=20=10*2

Crossrefs

Cf. A205558, A204892, A204890, A205725.

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 = 10; t = d[c]               (* A205718 *)
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}]        (* A205720 *)
Table[j[n], {n, 1, z2}]        (* A205721 *)
Table[s[k[n]], {n, 1, z2}]     (* A205722 *)
Table[s[j[n]], {n, 1, z2}]     (* A205723 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205724 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205725 *)

A205560 Numbers k for which 3 divides prime(k)-prime(j) for some j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(3)-p(1)=5-2=3=3*1
p(5)-p(1)=11-2=9=3*3
p(5)-p(3)=11-5=6=3*2
p(6)-p(4)=13-7=6=3*2
p(7)-p(1)=17-2=15=3*5
p(7)-p(3)=17-5=12=3*4

Links

Crossrefs

Cf. A204892, A205547, A204890, A205675.

Programs

  • Maple
    R:= NULL: N[0]:= 0: N[1]:= 0: N[2]:= 0: p:= 0:
for k from 1 to 30 do
  p:= nextprime(p);
  v:= p mod 3;
  R:= R, k$N[v];
  N[v]:= N[v]+1;
od:
R; # Robert Israel, Nov 18 2024
  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80;
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 = 3; t = d[c]              (* A205559 *)
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}]        (* A205560 *)
Table[j[n], {n, 1, z2}]        (* A205547 *)
Table[s[k[n]], {n, 1, z2}]     (* A205673 *)
Table[s[j[n]], {n, 1, z2}]     (* A205674 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205557 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205675 *)

A205677 Numbers k for which 4 divides prime(k)-prime(j) for some j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(4)-p(2)=7-3=4=4*1
p(5)-p(2)=11-3=8=4*2
p(5)-p(4)=11-7=4=4*1
p(6)-p(3)=13-5=8=4*2
p(7)-p(3)=17-5=12=4*3
p(7)-p(6)=17-13=4=4*1

Crossrefs

Cf. A204892, A205676, A204890, A205682.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80;
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 = 4; t = d[c]                (* A205676 *)
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}]         (* A205677 *)
Table[j[n], {n, 1, z2}]         (* A205678 *)
Table[s[k[n]], {n, 1, z2}]      (* A205679 *)
Table[s[j[n]], {n, 1, z2}]      (* A205680 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205681 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205682 *)

A205684 Numbers k for which 5 divides prime(k)-prime(j) for some j

Original entry on oeis.org

4, 6, 7, 7, 9, 9, 10, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(4)-p(1)=7-2=5=5*1
p(6)-p(2)=13-3=10=5*2
p(7)-p(1)=17-2=15=5*3
p(7)-p(4)=17-7=10=5*2
p(9)-p(2)=23-3=20=5*4
p(9)-p(6)=23-13=10=5*2

Crossrefs

Cf. A205558, A204892, A204890, A205685, A205689.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 80;
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 = 5; t = d[c]                (* A205683 *)
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}]        (* A205684 *)
Table[j[n], {n, 1, z2}]        (* A205685 *)
Table[s[k[n]], {n, 1, z2}]     (* A205686 *)
Table[s[j[n]], {n, 1, z2}]     (* A205687 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205688 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205689 *)

A205691 Numbers k for which 6 divides prime(k)-prime(j) for some j

Original entry on oeis.org

5, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20
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(3)=11-5=6=6*1
p(6)-p(4)=13-7=6=6*1
p(7)-p(3)=17-5=12=6*2
p(7)-p(5)=17-11=6=6*1
p(8)-p(4)=19-7=12=6*2
p(8)-p(6)=19-13=6=6*1

Crossrefs

Cf. A205558, A204892, A204890, A205695, A205696.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 400; z2 = 80;
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 = 6; t = d[c]                (* A205690 *)
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}]        (* A205691 *)
Table[j[n], {n, 1, z2}]        (* A205692 *)
Table[s[k[n]], {n, 1, z2}]     (* A205693 *)
Table[s[j[n]], {n, 1, z2}]     (* A205694 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205695 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205696 *)

A205698 Numbers k for which 7 divides prime(k)-prime(j) for some j

Original entry on oeis.org

7, 8, 9, 11, 11, 12, 12, 13, 14, 15, 15, 16, 17, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 24, 24, 24, 24, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32
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(7)-p(2)=17-3=14=7*2
p(8)-p(3)=19-5=14=7*2
p(9)-p(1)=23-2=21=7*3
p(11)-p(2)=31-3=28=7*4
p(11)-p(7)=31-17=14=7*2
p(12)-p(1)=37-2=35=7*5

Crossrefs

Cf. A205558, A204892, A204890, A205699, A205703.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]; z1 = 1200; z2 = 80;
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 = 7; t = d[c]                (* A205697 *)
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}]         (* A205698 *)
Table[j[n], {n, 1, z2}]         (* A205699 *)
Table[s[k[n]], {n, 1, z2}]      (* A205700 *)
Table[s[j[n]], {n, 1, z2}]      (* A205701 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205702 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205703 *)

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 *)

A205678 The number j such that 4 divides prime(k)-prime(j), where k(n)=A205677(n).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205677.)

Crossrefs

Cf. A205677, A205558.

Programs

  • Mathematica
    (See the program at A205677.)

A205685 The number j such that 5 divides prime(k)-prime(j), where k(n)=A205684(n).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205684.)

Crossrefs

Cf. A205684, A205558.

Programs

  • Mathematica
    (See the program at A205684.)
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