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 A204924 Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=A000045(k+1) (Fibonacci numbers). 10

%I

%S 2,3,4,4,5,5,5,6,7,6,6,6,7,9,12,7,9,7,7,7,8,10,12,12,9,8,9,10,8,17,8,

%T 8,8,9,21,12,14,10,18,17,11,9,10,10,12,12,9,12,13,9,17,9,9,9,10,17,12,

%U 12,25,22

%N Least k such that n divides s(k)-s(j) for some j in [1,k), where s(k)=A000045(k+1) (Fibonacci numbers).

%C See A204892 for a discussion and guide to related sequences.

%e 1 divides s(2)-s(1), so a(1)=2

%e 2 divides s(3)-s(1), so a(2)=3

%e 3 divides s(4)-s(2), so a(3)=4

%e 9 divides s(7)-s(3), so a(9)=7

%t s[n_] := s[n] = Fibonacci[n + 1]; z1 = 300; z2 = 60;

%t Table[s[n], {n, 1, 30}]

%t u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

%t Table[u[m], {m, 1, z1}] (* A204922 *)

%t v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

%t w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

%t d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]

%t Table[d[n], {n, 1, z2}] (* A204923 *)

%t k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]

%t m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]

%t j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2

%t Table[k[n], {n, 1, z2}] (* A204924 *)

%t Table[j[n], {n, 1, z2}] (* A204925 *)

%t Table[s[k[n]], {n, 1, z2}] (* A204926 *)

%t Table[s[j[n]], {n, 1, z2}] (* A204927 *)

%t Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A204928 *)

%t Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204929 *)

%Y Cf. A000045, A204892.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jan 21 2012

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