login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A205558 (A204898)/2 = (prime(k)-prime(j))/2; A086802 without its zeros. 58
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

Table of n, a(n) for n=1..80.

EXAMPLE

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

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 *)

CROSSREFS

Cf. A205675, A205560, A204892.

Sequence in context: A087850 A087849 A075015 * A082494 A194187 A174375

Adjacent sequences:  A205555 A205556 A205557 * A205559 A205560 A205561

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 30 2012

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 20 14:06 EDT 2019. Contains 326152 sequences. (Running on oeis4.)