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 1-3 of 3 results.

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 *)

A204899 Least k such that n divides A204898(k), the k-th difference of two odd primes.

Original entry on oeis.org

1, 1, 5, 2, 7, 5, 11, 4, 23, 7, 31, 12, 29, 11, 48, 16, 46, 23, 56, 22, 80, 31, 94, 30, 92, 29, 107, 37, 121, 48, 138, 47, 155, 46, 172, 57, 192, 56, 212, 67, 234, 80, 232, 79, 256, 94, 254, 93, 277, 92
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2012

Keywords

Comments

For a guide to related sequences, see A204892.

Crossrefs

Cf. A204900, A204892.

Programs

  • Mathematica
    (See the program at A204900.)

A204900 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k) is the k-th odd prime.

Original entry on oeis.org

2, 2, 4, 3, 5, 4, 6, 4, 8, 5, 9, 6, 9, 6, 11, 7, 11, 8, 12, 8, 14, 9, 15, 9, 15, 9, 16, 10, 17, 11, 18, 11, 19, 11, 20, 12, 21, 12, 22, 13, 23, 14, 23, 14, 24, 15, 24, 15, 25, 15, 27, 16, 28, 16, 29, 16, 30, 17, 31, 18, 30, 18, 31, 18, 32, 19, 32, 19, 34, 20, 34, 21, 34
Offset: 1

Views

Author

Clark Kimberling, Jan 20 2012

Keywords

Comments

See A204892 for a discussion and guide to related sequences

Crossrefs

Cf. A000040, A065091, A204892.

Programs

  • Mathematica
    s[n_] := s[n] = Prime[n + 1]; z1 = 400; z2 = 50;
Table[s[n], {n, 1, 30}]      (* A065091 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]      (* A204898 *)
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}]      (* A204899 *)
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}]      (* A204900 *)
Table[j[n], {n, 1, z2}]      (* A204901 *)
Table[s[k[n]], {n, 1, z2}]   (* A204902 *)
Table[s[j[n]], {n, 1, z2}]   (* A204903 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204904 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A000034 conjectured *)
Showing 1-3 of 3 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.