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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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 *)

A204994 Least k such that n divides A120070(k+1), the k-th difference between distinct squares.

Original entry on oeis.org

1, 2, 1, 2, 3, 5, 6, 2, 10, 14, 15, 5, 21, 27, 4, 9, 36, 31, 45, 14, 8, 65, 66, 7, 41, 90, 13, 27, 105, 23, 120, 12, 19, 152, 11, 31, 171, 189, 26, 18, 210, 40, 231, 65, 17, 275, 276, 16, 85, 96, 43, 90, 351, 61, 24, 33, 53, 434, 435, 23
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204994, A204892.

Programs

  • Mathematica
    (See the program at A204994.)

A204916 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=(prime(k))^2.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 20 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences

Crossrefs

Cf. A000040, A204892, A204900, A204908.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n]^2; z1 = 1000; z2 = 60;
Table[s[n], {n, 1, 30}]  (* A001248 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]             (* A204914 *)
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] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]             (* A204915 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]             (* A204916 *)
Table[j[n], {n, 1, z2}]             (* A204917 *)
Table[s[k[n]], {n, 1, z2}]          (* A204918 *)
Table[s[j[n]], {n, 1, z2}]          (* A204919 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204920 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204921 *)

A204979 Least k such that n divides 2^(k-1)-2^(j-1) for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 3, 4, 5, 4, 4, 5, 7, 6, 11, 5, 13, 5, 5, 6, 9, 8, 19, 7, 7, 12, 12, 6, 21, 14, 19, 6, 29, 6, 6, 7, 11, 10, 13, 9, 37, 20, 13, 8, 21, 8, 15, 13, 13, 13, 24, 7, 22, 22
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Examples

			1 divides 2^2-2^1, so a(1)=2
2 divides 2^3-2^2, so a(2)=3
3 divides 2^3-2^1, so a(3)=3
4 divides 2^4-2^3, so a(4)=4

Crossrefs

Cf. A000079, A204892.

Programs

  • Mathematica
    s[n_] := s[n] = 2^(n - 1); z1 = 800; z2 = 50;
Table[s[n], {n, 1, 30}]       (* A000079 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]       (* A130328 *)
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] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]       (* A204939 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A204979 *)
Table[j[n], {n, 1, z2}]       (* A001511 ? *)
Table[s[k[n]], {n, 1, z2}]    (* A204981 *)
Table[s[j[n]], {n, 1, z2}]    (* A006519 ? *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204983 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204984 *)

A205007 a(n) = (1/n)*A205006(n), where A205006(n) = s(k)-s(j), with (s(k),s(j)) the least pair of distinct triangular numbers for which n divides their difference.

Original entry on oeis.org

2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Links

Crossrefs

Cf. A000217, A205002, A205006, A204892.
Cf. A318894 (gives the positions terms larger than one).

Programs

  • Mathematica
    (See the program at A205002.)
  • PARI
    A205007(n) = for(k=2,oo,my(sk=binomial(k+1,2)); for(j=1,k-1,if(!((sk-binomial(j+1,2))%n),return((sk-binomial(j+1,2))/n)))); \\ Antti Karttunen, Sep 27 2018

Extensions

More terms from Antti Karttunen, Sep 27 2018

A205842 Numbers k for which 3 divides s(k)-s(j) for some j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

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

Crossrefs

Cf. A204892, 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 *)

A205847 Numbers k for which 4 divides s(k)-s(j) for some j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(4)-s(1) = 5-1 = 4
s(6)-s(1) = 13-1 = 12
s(6)-s(4) = 13-5 = 8
s(7)-s(1) = 21-1 = 20
s(7)-s(4) = 21-5 = 16
s(7)-s(6) = 21-13 = 8

Crossrefs

Cf. A204892, A205850.

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 = 4; t = d[c]    (* A205846 *)
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}]    (* A205847 *)
Table[j[n], {n, 1, z2}]    (* A205848 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)
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