A205712 Numbers k for which 9 divides prime(k)-prime(j) for some j<k; each k occurs once for each such 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

1,1

Comments

For a guide to related sequences, see A205558.

Links

Table of n, a(n) for n=1..67.

Example

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

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 *)

Crossrefs

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

Sequence in context: A068388 A314582 A314583 * A093711 A325204 A163670

Adjacent sequences: A205709 A205710 A205711 * A205713 A205714 A205715

Keyword

nonn

Author

Clark Kimberling, Jan 31 2012

Status

approved