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

%I A205146 #10 Jul 23 2021 09:00:02
%S A205146 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,
%T A205146 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,
%U A205146 5,14,5,8,10,20,16,4,6,18,12,14,13,7,6,9,11
%N 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).
%C A205146 See A204892 for a discussion and guide to related sequences.
%H A205146 Michel Marcus, <a href="/A205146/b205146.txt">Table of n, a(n) for n = 1..10000</a>
%t A205146 s[n_] := s[n] = Prime[n] Prime[n + 1]; z1 = 400; z2 = 60;
%t A205146 Table[s[n], {n, 1, 30}]           (* A006094 *)
%t A205146 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A205146 Table[u[m], {m, 1, z1}]           (* A205144 *)
%t A205146 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A205146 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A205146 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A205146 Table[d[n], {n, 1, z2}]           (* A205145 *)
%t A205146 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A205146 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A205146 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A205146 Table[k[n], {n, 1, z2}]           (* A205146 *)
%t A205146 Table[j[n], {n, 1, z2}]           (* A205147 *)
%t A205146 Table[s[k[n]], {n, 1, z2}]        (* A205148 *)
%t A205146 Table[s[j[n]], {n, 1, z2}]        (* A205149 *)
%t A205146 Table[s[k[n]] - s[j[n]], {n, 1, z2}]        (* A205150 *)
%t A205146 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}]    (* A205151 *)
%o A205146 (PARI) s(m) = prime(m)*prime(m+1);
%o A205146 isok(k, n) = my(sk=s(k)); for (j=1, k-1, if (!Mod(sk-s(j), n), return (k)));
%o A205146 a(n) = my(k=1); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Jul 23 2021
%Y A205146 Cf. A006094, A204892, A205143.
%K A205146 nonn
%O A205146 1,1
%A A205146 _Clark Kimberling_, Jan 25 2012
%E A205146 More terms from _Michel Marcus_, Jul 23 2021