A205712 Numbers k for which 9 divides prime(k)-prime(j) for some j

5, 9, 10, 10, 11, 12, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 28, 29, 29, 29, 30, 30, 30, 30, 31, 31, 31, 31, 32, 32, 32, 32, 32, 33

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(1)=11-2=9=9*1
p(9)-p(3)=23-5=18=9*2
p(10)-p(1)=29-2=27=9*3
p(10)-p(5)=29-11=18=9*2
p(11)-p(6)=31-13=18=9*2
p(12)-p(8)=37-19=18=9*2
		

Crossrefs

Cf. A205558, A204892, A204890, A205713, A205717.

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 = 9; t = d[c]                (* A205711 *)
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}]        (* A205712 *)
Table[j[n], {n, 1, z2}]        (* A205713 *)
Table[s[k[n]], {n, 1, z2}]     (* A205714 *)
Table[s[j[n]], {n, 1, z2}]     (* A205715 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205716 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205717 *)