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A205691 Numbers k for which 6 divides prime(k)-prime(j) for some j<k; each k occurs once for each such j. 8
5, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20, 20 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For a guide to related sequences, see A205558.

LINKS

Table of n, a(n) for n=1..69.

EXAMPLE

The first six terms match these differences:

p(5)-p(3)=11-5=6=6*1

p(6)-p(4)=13-7=6=6*1

p(7)-p(3)=17-5=12=6*2

p(7)-p(5)=17-11=6=6*1

p(8)-p(4)=19-7=12=6*2

p(8)-p(6)=19-13=6=6*1

MATHEMATICA

s[n_] := s[n] = Prime[n]; z1 = 400; 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 = 6; t = d[c]                (* A205690 *)

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}]        (* A205691 *)

Table[j[n], {n, 1, z2}]        (* A205692 *)

Table[s[k[n]], {n, 1, z2}]     (* A205693 *)

Table[s[j[n]], {n, 1, z2}]     (* A205694 *)

Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A205695 *)

Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205696 *)

CROSSREFS

Cf. A205558, A204892, A204890, A205695, A205696.

Sequence in context: A095942 A139395 A029911 * A002141 A114547 A217293

Adjacent sequences:  A205688 A205689 A205690 * A205692 A205693 A205694

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jan 31 2012

STATUS

approved

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Last modified May 20 12:33 EDT 2019. Contains 323422 sequences. (Running on oeis4.)