A205560
Numbers k for which 3 divides prime(k)-prime(j) for some j
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, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19
Offset: 1
Keywords
Examples
The first six terms match these differences: p(3)-p(1)=5-2=3=3*1 p(5)-p(1)=11-2=9=3*3 p(5)-p(3)=11-5=6=3*2 p(6)-p(4)=13-7=6=3*2 p(7)-p(1)=17-2=15=3*5 p(7)-p(3)=17-5=12=3*4
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
R:= NULL: N[0]:= 0: N[1]:= 0: N[2]:= 0: p:= 0: for k from 1 to 30 do p:= nextprime(p); v:= p mod 3; R:= R, k$N[v]; N[v]:= N[v]+1; od: R; # Robert Israel, Nov 18 2024
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Mathematica
s[n_] := s[n] = Prime[n]; z1 = 200; z2 = 80; 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 = 3; t = d[c] (* A205559 *) 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}] (* A205560 *) Table[j[n], {n, 1, z2}] (* A205547 *) Table[s[k[n]], {n, 1, z2}] (* A205673 *) Table[s[j[n]], {n, 1, z2}] (* A205674 *) Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205557 *) Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205675 *)
Comments