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A205560 Numbers k for which 3 divides prime(k)-prime(j) for some j each k occurs once for each such j.

%I A205560 #14 Nov 19 2024 00:51:14
%S A205560 3,5,5,6,7,7,7,8,8,9,9,9,9,10,10,10,10,10,11,11,11,12,12,12,12,13,13,
%T A205560 13,13,13,13,14,14,14,14,14,15,15,15,15,15,15,15,16,16,16,16,16,16,16,
%U A205560 16,17,17,17,17,17,17,17,17,17,18,18,18,18,18,18,19,19,19,19
%N A205560 Numbers k for which 3 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
%C A205560 For a guide to related sequences, see A205558.
%H A205560 Robert Israel, <a href="/A205560/b205560.txt">Table of n, a(n) for n = 1..10000</a>
%e A205560 The first six terms match these differences:
%e A205560 p(3)-p(1)=5-2=3=3*1
%e A205560 p(5)-p(1)=11-2=9=3*3
%e A205560 p(5)-p(3)=11-5=6=3*2
%e A205560 p(6)-p(4)=13-7=6=3*2
%e A205560 p(7)-p(1)=17-2=15=3*5
%e A205560 p(7)-p(3)=17-5=12=3*4
%p A205560 R:= NULL: N[0]:= 0: N[1]:= 0: N[2]:= 0: p:= 0:
%p A205560 for k from 1 to 30 do
%p A205560   p:= nextprime(p);
%p A205560   v:= p mod 3;
%p A205560   R:= R, k$N[v];
%p A205560   N[v]:= N[v]+1;
%p A205560 od:
%p A205560 R; # _Robert Israel_, Nov 18 2024
%t A205560 s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80;
%t A205560 f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
%t A205560 Table[s[n], {n, 1, 30}]      (* A000040 *)
%t A205560 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205560 Table[u[m], {m, 1, z1}]      (* A204890 *)
%t A205560 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205560 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205560 d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
%t A205560 c = 3; t = d[c]              (* A205559 *)
%t A205560 k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
%t A205560 j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
%t A205560 Table[k[n], {n, 1, z2}]        (* A205560 *)
%t A205560 Table[j[n], {n, 1, z2}]        (* A205547 *)
%t A205560 Table[s[k[n]], {n, 1, z2}]     (* A205673 *)
%t A205560 Table[s[j[n]], {n, 1, z2}]     (* A205674 *)
%t A205560 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205557 *)
%t A205560 Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205675 *)
%Y A205560 Cf. A204892, A205547, A204890, A205675.
%K A205560 nonn
%O A205560 1,1
%A A205560 _Clark Kimberling_, Jan 30 2012