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Numbers k for which 9 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
8

%I #12 Oct 25 2024 04:52:06

%S 5,9,10,10,11,12,13,13,14,15,15,15,16,17,17,17,18,18,19,19,20,20,21,

%T 21,22,22,22,23,23,23,23,24,24,24,25,25,25,25,26,26,26,26,26,27,27,27,

%U 28,28,28,28,29,29,29,30,30,30,30,31,31,31,31,32,32,32,32,32,33

%N Numbers k for which 9 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(5)-p(1)=11-2=9=9*1

%e p(9)-p(3)=23-5=18=9*2

%e p(10)-p(1)=29-2=27=9*3

%e p(10)-p(5)=29-11=18=9*2

%e p(11)-p(6)=31-13=18=9*2

%e p(12)-p(8)=37-19=18=9*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 = 9; t = d[c] (* A205711 *)

%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}] (* A205712 *)

%t Table[j[n], {n, 1, z2}] (* A205713 *)

%t Table[s[k[n]], {n, 1, z2}] (* A205714 *)

%t Table[s[j[n]], {n, 1, z2}] (* A205715 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205716 *)

%t Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205717 *)

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

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 31 2012