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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A205852 Numbers k for which 5 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 6, 6, 7, 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 27
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(5)-s(3) = 8-3 = 5
s(6)-s(3) = 13-3 = 10
s(6)-s(5) = 13-8 = 5
s(7)-s(1) = 21-1 = 20
s(9)-s(4) = 55-5 = 50
s(10)-s(8) = 89-34 = 55

Crossrefs

Cf. A204892, A205855.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 5; t = d[c]    (* A205851 *)
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}] (* A205852 *)
Table[j[n], {n, 1, z2}] (* A205853 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}](* A205854 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205855 *)

A205857 Numbers k for which 6 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 6, 7, 9, 9, 10, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 28, 28, 28, 28
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(5)-s(2) = 8-2 = 6 = 6*1
s(6)-s(1) = 13-1 = 12 = 6*2
s(7)-s(3) = 21-3 = 18 = 6*3
s(9)-s(1) = 55-1 = 54 = 6*9
s(9)-s(6) = 55-13 = 42 = 6*7
s(10)-s(4) = 89-5 = 84 =6*14

Crossrefs

Cf. A204892, A205857, A205860.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 6; t = d[c]    (* A205856 *)
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}]     (* A205857 *)
Table[j[n], {n, 1, z2}]     (* A205858 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}]    (* A205859 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205860 *)

A205862 Numbers k for which 7 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 8, 9, 9, 10, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29
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(5)-s(1) = 8-1 = 7 = 7*1
s(8)-s(6) = 34-13 = 21 = 7*3
s(9)-s(6) = 55-13 = 42 = 7*6
s(9)-s(8) = 55-34 = 21 = 7*3
s(10)-s(4) = 89-5 = 84 = 7*12
s(13)-s(6) = 377-13 = 364 =7*52

Crossrefs

Cf. A204892, A205863, A205865.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; 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 = 7; t = d[c]   (* A205861 *)
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}]    (* A205862 *)
Table[j[n], {n, 1, z2}]    (* A205863 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}] (* A205865 *)

A205867 Numbers k for which 8 divides s(k)-s(j) for some j

Original entry on oeis.org

6, 7, 7, 8, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25
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(6)-s(4) = 13-5 = 8 = 8*1
s(7)-s(4) = 21-5 = 16 = 8*2
s(7)-s(6) = 21-13 = 8 = 8*1
s(8)-s(2) = 34-2 = 32 = 8*4
s(10)-s(1) = 89-1 = 88 = 8*11
s(11)-s(5) = 144-8 = 136 =8*17

Crossrefs

Cf. A204892, A205868, A205870.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
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 = 8; t = d[c]    (* A205866 *)
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}]      (* A205867 *)
Table[j[n], {n, 1, z2}]        (* A205868 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205870 *)

A205872 Numbers k for which 9 divides s(k)-s(j) for some j

Original entry on oeis.org

7, 9, 10, 12, 12, 13, 13, 13, 14, 16, 17, 17, 18, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 29, 30, 30, 31, 31, 31, 32, 32
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(7)-s(3) = 21-3 = 18 = 9*2
s(9)-s(1) = 55-1 = 54 = 9*6
s(10)-s(5) = 89-8 = 81 = 9*9
s(12)-s(5) = 233-8 = 225 = 9*25
s(12)-s(10) = 233-89 = 144 = 9*16
s(13)-s(5) = 377-8 = 369 =9*41

Crossrefs

Cf. A204892, A205873, A205875.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
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 = 9; t = d[c]     (* A205871 *)
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}]       (* A205872 *)
Table[j[n], {n, 1, z2}]         (* A205873 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]   (* A205874 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205875 *)

A205877 Numbers k for which 10 divides s(k)-s(j) for some j

Original entry on oeis.org

6, 7, 9, 11, 12, 12, 15, 16, 16, 17, 17, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 33, 34, 34
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first three terms match these differences:
s(6)-s(3) = 13-3 = 10 = 10*1
s(7)-s(1) = 21-1 = 20 = 10*2
s(9)-s(4) = 55-5 = 50 = 10*5

Crossrefs

Cf. A204892, A205878, A205880.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
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 = 10; t = d[c]    (* A205876 *)
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}]     (* A205877 *)
Table[j[n], {n, 1, z2}]      (* A205878 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205879 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205880 *)

A204911 The prime q>=5 such that n divides p-q, where p>q is the least prime for which such a prime q exists.

Original entry on oeis.org

5, 5, 5, 7, 7, 5, 5, 5, 5, 7, 7, 5, 5, 5, 7, 7, 7, 5, 5, 11, 5, 7, 7, 5, 11, 5, 5, 13, 13, 7, 5, 5, 5, 7, 13, 5, 5, 5, 5, 7, 7, 5, 11, 17, 7, 7, 7, 5, 5, 11, 5, 7, 7, 5, 17, 5, 13, 13, 13, 7, 5, 5, 5, 7, 7, 5, 5, 5, 11, 13, 7, 7, 5, 5, 7, 7, 13, 5, 5, 17, 5, 7, 7, 5, 11, 11, 5, 13, 13, 7, 11, 5, 5
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Links

Crossrefs

Cf. A204908, A204900, A204892.

Programs

  • Maple
    f:= proc(n) local V,q,r;
  V:= Array(0..n-1); q:= 4;
  do
   q:= nextprime(q);
   r:= q mod n;
   if V[r] = 0 then V[r]:= q
   else return V[r]
   fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jul 24 2018
  • Mathematica
    (See the program at A204908.)
  • Python
    from sympy import nextprime
def a(n):
    V, q = [0 for _ in range(n)], 4
    while True:
        q = nextprime(q)
        r = q%n
        if V[r] == 0: V[r] = q
        else: return int(V[r])
print([a(n) for n in range(1, 94)]) # Michael S. Branicky, Jun 25 2024 after Robert Israel

Extensions

More terms from Robert G. Wilson v, Jul 24 2018

A204930 Ordered differences of factorials.

Original entry on oeis.org

1, 5, 4, 23, 22, 18, 119, 118, 114, 96, 719, 718, 714, 696, 600, 5039, 5038, 5034, 5016, 4920, 4320, 40319, 40318, 40314, 40296, 40200, 39600, 35280, 362879, 362878, 362874, 362856, 362760, 362160, 357840, 322560, 3628799, 3628798
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Examples

			a(1)=s(2)-s(1)=2!-1!=1
a(2)=s(3)-s(1)=3!-1!=5
a(3)=s(3)-s(2)=3!-2!=4
a(4)=s(4)-s(1)=4!-1!=23

Crossrefs

Cf. A204932, A204892.

Programs

  • Mathematica
    (See the program at A204932.)

A204991 2^k-2^j, where (2^k,2^j) is the least pair of distinct positive powers of 2 for which n divides 2^k-2^j.

Original entry on oeis.org

2, 2, 6, 4, 30, 6, 14, 8, 126, 30, 2046, 12, 8190, 14, 30, 16, 510, 126, 524286, 60, 126, 2046, 4094, 24, 2097150, 8190, 524286, 28, 536870910, 30, 62, 32, 2046, 510, 8190, 252, 137438953470, 524286, 8190, 120, 2097150, 126, 32766, 4092, 8190
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204987, A204892, A204990=(1/2)(A204991).

Programs

  • Mathematica
    (See the program at A204987.)

A204999 a(n) = (1/n)*A204998(n).

Original entry on oeis.org

3, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Clark Kimberling, Jan 21 2012

Keywords

Comments

For a guide to related sequences, see A204892.
Positions of 3's seem to be given by a subsequence of A104777. - Antti Karttunen, Sep 29 2018

Links

Crossrefs

Cf. A204996, A204997, A204998, A204892.

Programs

  • Mathematica
    (See the program at A204994.)
  • PARI
    A204999(n) = { my(d); for(k=sqrtint(1+n), oo, for(j=1,k-1,if(!((d=(k^2)-(j^2))%n),return(d/n),if(dAntti Karttunen, Sep 28 2018

Formula

a(n) = (A204996(n)-A204997(n))/n.

Extensions

More terms from Antti Karttunen, Sep 28 2018
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