A205146 Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=prime(j)*prime(j+1).

2, 3, 2, 3, 3, 4, 4, 5, 2, 3, 5, 5, 6, 4, 7, 5, 7, 5, 8, 3, 4, 5, 9, 6, 12, 6, 5, 7, 3, 7, 4, 5, 5, 7, 15, 5, 12, 8, 6, 8, 7, 4, 6, 7, 7, 9, 10, 6, 8, 12, 7, 10, 16, 5, 16, 13, 8, 10, 9, 7, 16, 4, 10, 5, 14, 5, 8, 10, 20, 16, 4, 6, 18, 12, 14, 13, 7, 6, 9, 11

Offset

1,1

Comments

See A204892 for a discussion and guide to related sequences.

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Mathematica

s[n_] := s[n] = Prime[n] Prime[n + 1]; z1 = 400; z2 = 60;
Table[s[n], {n, 1, 30}]           (* A006094 *)
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]           (* A205144 *)
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] = First[Delete[w[n], Position[w[n], 0]]]
Table[d[n], {n, 1, z2}]           (* A205145 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
Table[k[n], {n, 1, z2}]           (* A205146 *)
Table[j[n], {n, 1, z2}]           (* A205147 *)
Table[s[k[n]], {n, 1, z2}]        (* A205148 *)
Table[s[j[n]], {n, 1, z2}]        (* A205149 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]        (* A205150 *)
Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]    (* A205151 *)

Prog

(PARI) s(m) = prime(m)*prime(m+1);
isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (k)));
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jul 23 2021

Crossrefs

Cf. A006094, A204892, A205143.

Sequence in context: A082597 A112212 A102314 * A031248 A030582 A217438

Adjacent sequences: A205143 A205144 A205145 * A205147 A205148 A205149

Keyword

nonn

Author

Clark Kimberling, Jan 25 2012

Extensions

More terms from Michel Marcus, Jul 23 2021

Status

approved