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

1,1

Comments

For a guide to related sequences, see A205558.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

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

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

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 *)

Crossrefs

Cf. A204892, A205547, A204890, A205675.

Sequence in context: A029912 A272756 A322350 * A195939 A331859 A317587

Adjacent sequences: A205557 A205558 A205559 * A205561 A205562 A205563

Keyword

nonn

Author

Clark Kimberling, Jan 30 2012

Status

approved