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 1-8 of 8 results.

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.)

A205725 (prime(k)-prime(j))/10, where the pairs (k,j) are given by A205720 and A205721.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 3, 2, 3, 1, 4, 3, 2, 4, 3, 1, 5, 4, 3, 1, 4, 3, 5, 3, 2, 6, 5, 3, 2, 6, 4, 3, 1, 7, 6, 5, 3, 2, 6, 5, 2, 8, 7, 6, 4, 3, 1, 7, 6, 3, 1, 9, 8, 6, 5, 3, 9, 7, 6, 4, 3, 10, 9, 8, 6, 5, 3, 2, 10, 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 A205720.)

Crossrefs

Cf. A204890, A205720, A205558, A204724.

Programs

  • Mathematica
    (See the program at A205720.)

A205838 The least number j such that 2 divides s(k)-s(j), where k(n)=A205720(n).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205587.)

Crossrefs

Cf. A204890, A205587, A205840.

Programs

  • Mathematica
    (See the program at A205587.)

A205724 prime(k)-prime(j), where the pairs (k,j) are given by A205720 and A205721.

Original entry on oeis.org

10, 10, 20, 10, 10, 20, 30, 20, 30, 10, 40, 30, 20, 40, 30, 10, 50, 40, 30, 10, 40, 30, 50, 30, 20, 60, 50, 30, 20, 60, 40, 30, 10, 70, 60, 50, 30, 20, 60, 50, 20, 80, 70, 60, 40, 30, 10, 70, 60, 30, 10, 90, 80, 60, 50, 30, 90, 70, 60, 40, 30, 100, 90, 80, 60, 50
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, A205725.

Programs

  • Mathematica
    (See the program at A205720.)

A205722 Prime(A205720(n)), the n-th number s(k) such that 10 divides s(k)-s(j) for some j

Original entry on oeis.org

13, 17, 23, 23, 29, 31, 37, 37, 41, 41, 43, 43, 43, 47, 47, 47, 53, 53, 53, 53, 59, 59, 61, 61, 61, 67, 67, 67, 67, 71, 71, 71, 71, 73, 73, 73, 73, 73, 79, 79, 79, 83, 83, 83, 83, 83, 83, 89, 89, 89, 89, 97, 97, 97, 97, 97, 101, 101, 101, 101, 101, 103, 103
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.)

A205723 Prime(A205721(n)), the n-th number s(j) such that 10 divides s(k)-s(j), where the pairs (k,j) are given by A205720 and A205721.

Original entry on oeis.org

3, 7, 3, 13, 19, 11, 7, 17, 11, 31, 3, 13, 23, 7, 17, 37, 3, 13, 23, 43, 19, 29, 11, 31, 41, 7, 17, 37, 47, 11, 31, 41, 61, 3, 13, 23, 43, 53, 19, 29, 59, 3, 13, 23, 43, 53, 73, 19, 29, 59, 79, 7, 17, 37, 47, 67, 11, 31, 41, 61, 71, 3, 13, 23, 43, 53, 73, 83, 7, 17
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.)

A205558 (A204898)/2 = (prime(k)-prime(j))/2; A086802 without its zeros.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

Let p(n) denote the n-th prime. If c is a positive integer, there are infinitely many pairs (k,j) such that c divides p(k)-p(j). The set of differences p(k)-p(j) is ordered as a sequence at A204890. Guide to related sequences:
c....k..........j..........p(k)-p(j).[p(k)-p(j)]/c
2....A133196....A131818....A204898....A205558
3....A205560....A205547....A205557....A205675
4....A205677....A205678....A205681....A205682
5....A205684....A205685....A205688....A205689
6....A205691....A205692....A205695....A205696
7....A205698....A205699....A205702....A205703
8....A205705....A205706....A205709....A205710
9....A205712....A205713....A205716....A205717
10...A205720....A205721....A205724....A205725
It appears that, as rectangular array, this sequence can be described by A(n,k) is the least m such that there are k primes in the set prime(n) + 2*i for {i=1..n}. - Michel Marcus, Mar 29 2023

Examples

			Writing prime(k) as p(k),
p(3)-p(2)=5-3=2
p(4)-p(2)=7-3=4
p(4)-p(3)=7-5=2
p(5)-p(2)=11-3=8
p(5)-p(3)=11-5=6
p(5)-p(4)=11-7=4,
so that the first 6 terms of A205558 are 1,2,1,4,3,2.
The sequence can be regarded as a rectangular array in which row n is given by [prime(n+2+k)-prime(n+1)]/2; a northwest corner follows:
1...2...4...5...7...8....10...13...14...17...19...20
1...3...4...6...7...9....12...13...16...18...19...21
2...3...5...6...8...11...12...15...17...18...20...23
1...3...4...6...9...10...13...15...16...18...21...24
2...3...5...8...9...12...14...15...17...20...23...24
1...3...6...7...10..12...13...15...18...21...22...25
2...5...6...9...11..12...14...17...20...21...24...26
- _Clark Kimberling_, Sep 29 2013

Crossrefs

Cf. A205675, A205560, A204892.

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 = 2; t = d[c]                      (* A080036 *)
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}]                  (* A133196 *)
Table[j[n], {n, 1, z2}]                  (* A131818 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204898 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205558 *)

A205718 Positions of multiples of 10 in A204890 (differences of primes).

Original entry on oeis.org

12, 19, 30, 34, 44, 50, 59, 62, 71, 77, 80, 84, 87, 95, 98, 103, 107, 111, 114, 119, 128, 130, 141, 147, 149, 157, 160, 165, 168, 176, 182, 184, 189, 192, 196, 199, 204, 206, 218, 220, 227, 233, 237, 240, 245, 247, 252, 261, 263, 270, 275, 280, 283
Offset: 1

Views

Author

Clark Kimberling, Jan 31 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			In A204890=(1,3,2,5,4,2,9,8,6,4,11,10,8,...), multiples of 10 are in positions 12,19,30,...   See the example at A205720.

Links

Crossrefs

Cf. A204890, A205720, A205558.

Programs

Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.