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.

A205557 prime(k)-prime(j), where the pairs (k,j) are given by A205560 and A205547.

Original entry on oeis.org

3, 9, 6, 6, 15, 12, 6, 12, 6, 21, 18, 12, 6, 27, 24, 18, 12, 6, 24, 18, 12, 30, 24, 18, 6, 39, 36, 30, 24, 18, 12, 36, 30, 24, 12, 6, 45, 42, 36, 30, 24, 18, 6, 51, 48, 42, 36, 30, 24, 12, 6, 57, 54, 48, 42, 36, 30, 18, 12, 6, 54, 48, 42, 30, 24, 18, 60, 54, 48, 36
Offset: 1

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			(See the example at A205560.)

Crossrefs

Cf. A205558, A205560, A205675.

Programs

  • Mathematica
    (See the program at A205560.)

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 *)

A205560 Numbers k for which 3 divides prime(k)-prime(j) for some j

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 30 2012

Keywords

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(3)-p(1)=5-2=3=3*1
p(5)-p(1)=11-2=9=3*3
p(5)-p(3)=11-5=6=3*2
p(6)-p(4)=13-7=6=3*2
p(7)-p(1)=17-2=15=3*5
p(7)-p(3)=17-5=12=3*4

Links

Crossrefs

Cf. A204892, A205547, A204890, A205675.

Programs

  • Maple
    R:= NULL: N[0]:= 0: N[1]:= 0: N[2]:= 0: p:= 0:
for k from 1 to 30 do
  p:= nextprime(p);
  v:= p mod 3;
  R:= R, k$N[v];
  N[v]:= N[v]+1;
od:
R; # Robert Israel, Nov 18 2024
  • 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 = 3; t = d[c]              (* A205559 *)
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}]        (* A205560 *)
Table[j[n], {n, 1, z2}]        (* A205547 *)
Table[s[k[n]], {n, 1, z2}]     (* A205673 *)
Table[s[j[n]], {n, 1, z2}]     (* A205674 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205557 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205675 *)
Showing 1-3 of 3 results.

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