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A205677
Numbers k for which 4 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j.
8
4, 5, 5, 6, 7, 7, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19
OFFSET
1,1
COMMENTS
For a guide to related sequences, see A205558.
EXAMPLE
The first six terms match these differences:
p(4)-p(2)=7-3=4=4*1
p(5)-p(2)=11-3=8=4*2
p(5)-p(4)=11-7=4=4*1
p(6)-p(3)=13-5=8=4*2
p(7)-p(3)=17-5=12=4*3
p(7)-p(6)=17-13=4=4*1
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 = 4; t = d[c] (* A205676 *)
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}] (* A205677 *)
Table[j[n], {n, 1, z2}] (* A205678 *)
Table[s[k[n]], {n, 1, z2}] (* A205679 *)
Table[s[j[n]], {n, 1, z2}] (* A205680 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205681 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205682 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jan 30 2012
STATUS
approved