A205720 - OEIS

%I #9 Mar 30 2012 18:58:11

%S 6,7,9,9,10,11,12,12,13,13,14,14,14,15,15,15,16,16,16,16,17,17,18,18,

%T 18,19,19,19,19,20,20,20,20,21,21,21,21,21,22,22,22,23,23,23,23,23,23,

%U 24,24,24,24,25,25,25,25,25,26,26,26,26,26,27,27,27,27,27,27

%N Numbers k for which 10 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.

%C For a guide to related sequences, see A205558.

%e The first six terms match these differences:

%e p(6)-p(2)=13-3=10=10*1

%e p(7)-p(4)=17-7=10=10*1

%e p(9)-p(2)=23-3=20=10*2

%e p(9)-p(6)=23-13=10=10*1

%e p(10)-p(8)=29-19=10=10*1

%e p(11)-p(5)=31-11=20=10*2

%t s[n_] := s[n] = Prime[n]; z1 = 900; z2 = 70;

%t f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];

%t Table[s[n], {n, 1, 30}]        (* A000040 *)

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}]        (* A204890 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]

%t c = 10; t = d[c]               (* A205718 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]

%t j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2

%t Table[k[n], {n, 1, z2}]        (* A205720 *)

%t Table[j[n], {n, 1, z2}]        (* A205721 *)

%t Table[s[k[n]], {n, 1, z2}]     (* A205722 *)

%t Table[s[j[n]], {n, 1, z2}]     (* A205723 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205724 *)

%t Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205725 *)

%Y Cf. A205558, A204892, A204890, A205725.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 31 2012