A205705 Numbers k for which 8 divides prime(k)-prime(j) for some j

5, 6, 8, 8, 9, 10, 10, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 22, 22, 22, 23, 23, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26

Offset: 1

Author

Clark Kimberling, Jan 31 2012

Keywords

• nonn

Comments

For a guide to related sequences, see A205558.

Examples

			The first six terms match these differences:
p(5)-p(2)=11-3=8=8*1
p(6)-p(3)=13-5=8=8*1
p(8)-p(2)=19-3=16=8*2
p(8)-p(5)=19-11=8=8*1
p(9)-p(4)=23-7=16=8*2
p(10)-p(3)=29-5=24=8*3
		

Crossrefs

Cf. A205558, A204892, A204890, A205706, A205710.

Programs

Mathematica

s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;
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 = 8; t = d[c]             (* A205704 *)
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}]         (* A205705 *)
Table[j[n], {n, 1, z2}]         (* A205706 *)
Table[s[k[n]], {n, 1, z2}]      (* A205707 *)
Table[s[j[n]], {n, 1, z2}]      (* A205708 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]      (* A205709 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205710 *)