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

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

Author

Clark Kimberling, Jan 30 2012

Keywords

• nonn

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 *)