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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A205000 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 3, 4, 3, 4, 6, 4, 7, 3, 4, 4, 6, 3, 6, 4, 5, 6, 4, 5, 6, 7, 9, 7, 13, 4, 9, 4, 6, 6, 5, 8, 9, 6, 8, 6, 7, 5, 12, 6, 8, 4, 8, 5, 7, 6, 7, 8, 16, 9, 6, 7, 9, 13, 13, 4
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A001047, A204892, A205111.

Programs

  • Mathematica
    s[n_] := s[n] = 3^n - 2^n; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]      (* A001047 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]      (* A205105 *)
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}]  (* A205106 *)
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}]                  (* A205000 *)
Table[j[n], {n, 1, z2}]                  (* A205107 *)
Table[s[k[n]], {n, 1, z2}]               (* A205108 *)
Table[s[j[n]], {n, 1, z2}]               (* A205109 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205110 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205111 *)

A205010 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 4, 3, 3, 4, 4, 5, 4, 5, 8, 7, 6, 4, 8, 5, 5, 4, 4, 8, 7, 8, 5, 7, 5, 6, 7, 7, 6, 8, 12, 5, 8, 5, 8, 10, 10, 8, 9, 9, 6, 7, 16, 8, 10, 11, 8, 7, 8, 5, 7, 10, 8, 7, 10, 7, 8, 6, 20, 8
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A000984, A204892, A205008.

Programs

  • Mathematica
    s[n_] := s[n] = Binomial[2 (n - 1), n - 1];
 z1 = 700; z2 = 60;
Table[s[n], {n, 1, 30}]   (* A000984 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A205008 *)
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}]   (* A205009 *)
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}]      (* A205010 *)
Table[j[n], {n, 1, z2}]      (* A205011 *)
Table[s[k[n]], {n, 1, z2}]   (* A205012 *)
Table[s[j[n]], {n, 1, z2}]   (* A205013 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205014 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205015 *)

A205018 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 2, 3, 2, 3, 3, 4, 4, 4, 3, 6, 5, 7, 4, 6, 8, 9, 4, 10, 6, 8, 6, 12, 5, 7, 7, 7, 5, 15, 6, 16, 16, 8, 9, 8, 6, 19, 10, 9, 6, 21, 8, 22, 7, 10, 12, 24, 9, 10, 7, 11, 8, 27, 7, 13, 11, 12, 15, 30, 8
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A002378, A204892, A205016.

Programs

  • Mathematica
    s[n_] := s[n] = n*(n + 1); z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]   (* A002378 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A205016 *)
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}]   (* A205017 *)
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}]      (* A205018 *)
Table[j[n], {n, 1, z2}]      (* A205028 *)
Table[s[k[n]], {n, 1, z2}]   (* A205029 *)
Table[s[j[n]], {n, 1, z2}]   (* A205030 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205031 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205032 *)

A205114 Least k such that n divides L(k)-L(j) for some j satisfying 1<=jA000032).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 22 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A000032, A204892, A205119.

Programs

  • Mathematica
    s[n_] := s[n] = LucasL[n]; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]    (* A000032 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]    (* A205112 *)
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}]     (* A205113 *)
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}]       (* A205114 *)
Table[j[n], {n, 1, z2}]       (* A205115 *)
Table[s[k[n]], {n, 1, z2}]    (* A205116 *)
Table[s[j[n]], {n, 1, z2}]    (* A205117 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205118 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205119 *)

A205122 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A001787, A204892, A205127.

Programs

  • Mathematica
    s[n_] := s[n] = n*2^(n - 1); z1 = 250; z2 = 70;
Table[s[n], {n, 1, 30}]   (* A001787 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A205120 *)
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}]   (* A205121 *)
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}]            (* A205122 *)
Table[j[n], {n, 1, z2}]            (* A205123 *)
Table[s[k[n]], {n, 1, z2}]         (* A205124 *)
Table[s[j[n]], {n, 1, z2}]         (* A205125 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205126 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205127 *)

A205130 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 3, 3, 5, 2, 5, 3, 9, 3, 5, 4, 6, 4, 3, 5, 17, 5, 7, 6, 6, 6, 4, 7, 11, 7, 11, 4, 11, 8, 5, 9, 33, 9, 14, 8, 9, 10, 6, 5, 13, 11, 12, 12, 5, 7, 7, 13, 18, 9, 14, 6, 12, 14, 8, 11, 13, 8, 23, 16, 6
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A000384, A204892, A205135.

Programs

  • Mathematica
    s[n_] := s[n] = 2 n^2 - n; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}] (* A000384, hexagonal numbers *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}] (* A205128 *)
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}]        (* A205129 *)
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}]        (* A205130 *)
Table[j[n], {n, 1, z2}]        (* A205131 *)
Table[s[k[n]], {n, 1, z2}]     (* A205132 *)
Table[s[j[n]], {n, 1, z2}]     (* A205133 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205134 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205135 *)

A205138 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

2, 2, 4, 2, 4, 5, 3, 6, 10, 4, 3, 7, 5, 8, 5, 6, 4, 10, 7, 8, 4, 8, 5, 7, 6, 13, 28, 9, 6, 5, 11, 22, 9, 5, 7, 10, 13, 18, 6, 8, 8, 11, 15, 15, 13, 6, 9, 7, 13, 6, 13, 15, 10, 29, 10, 9, 8, 7, 11, 15
Offset: 1

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A000326, A204892, A205143.

Programs

  • Mathematica
    s[n_] := s[n] = n (3 n - 1)/2; z1 = 500; z2 = 60;
Table[s[n], {n, 1, 30}]   (* A000326, pentagonal *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]    (* A205136 *)
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}]    (* A205137 *)
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}]        (* A205138 *)
Table[j[n], {n, 1, z2}]        (* A205139 *)
Table[s[k[n]], {n, 1, z2}]     (* A205140 *)
Table[s[j[n]], {n, 1, z2}]     (* A205141 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205142 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205143 *)

A205146 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 25 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Links

Crossrefs

Cf. A006094, A204892, A205143.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n] Prime[n + 1]; z1 = 400; z2 = 60;
Table[s[n], {n, 1, 30}]           (* A006094 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]           (* A205144 *)
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}]           (* A205145 *)
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}]           (* A205146 *)
Table[j[n], {n, 1, z2}]           (* A205147 *)
Table[s[k[n]], {n, 1, z2}]        (* A205148 *)
Table[s[j[n]], {n, 1, z2}]        (* A205149 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]        (* A205150 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]    (* A205151 *)
  • PARI
    s(m) = prime(m)*prime(m+1);
isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (k)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jul 23 2021

Extensions

More terms from Michel Marcus, Jul 23 2021

A205153 Least k such that n divides s(k)-s(j) for some j

Original entry on oeis.org

2, 2, 3, 4, 3, 4, 6, 4, 5, 8, 5, 6, 9, 6, 7, 10, 7, 9, 10, 8, 9, 8, 14, 9, 10, 9, 11, 10, 14, 10, 20, 14, 11, 16, 11, 12, 20, 12, 13, 22, 13, 16, 17, 14, 16, 14, 15, 16, 17, 15, 16, 15, 30, 16, 17, 16, 18, 17, 23, 17, 20, 31, 18, 20, 18, 19, 20, 19, 21, 20, 22, 20, 24
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A024675, A204892, A205375.

Programs

  • Mathematica
    s[n_] := s[n] = (Prime[n + 1] + Prime[n + 2])/2; z1 = 1100; z2 = 80;
Table[s[n], {n, 1, 30}]     (* A024675 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204980 *)
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}]     (* A205152 *)
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}]       (* A205153 *)
Table[j[n], {n, 1, z2}]       (* A205154 *)
Table[s[k[n]], {n, 1, z2}]    (* A205372 *)
Table[s[j[n]], {n, 1, z2}]    (* A205373 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205374 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205375 *)

A205378 Least k such that n divides s(k)-s(j) for some j

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 5, 2, 5, 4, 7, 3, 8, 5, 6, 3, 10, 5, 11, 4, 7, 7, 13, 3, 8, 8, 8, 5, 16, 6, 17, 5, 9, 10, 9, 5, 20, 11, 10, 4, 22, 7, 23, 7, 10, 13, 25, 4, 11, 8, 12, 8, 28, 8, 11, 5, 13, 16, 31, 6
Offset: 1

Views

Author

Clark Kimberling, Jan 26 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences.

Crossrefs

Cf. A016754, A204892, A205383.

Programs

  • Mathematica
    s[n_] := s[n] = (2 n - 1)^2; z1 = 600; z2 = 60;
Table[s[n], {n, 1, 30}]     (* A016754, odd squares *)
u[m_] :=  u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A205376 *)
%/8                         (* A049777 *)
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}]     (* A205377 *)
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}]     (* A205378 *)
Table[j[n], {n, 1, z2}]     (* A205379 *)
Table[s[k[n]], {n, 1, z2}]  (* A205380 *)
Table[s[j[n]], {n, 1, z2}]  (* A205381 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205382 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]  (* A205383 *)
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