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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).

%I A204924 #8 Mar 16 2022 20:12:00
%S A204924 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 A204924 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 A204924 12,25,22
%N 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).
%C A204924 See A204892 for a discussion and guide to related sequences.
%H A204924 G. C. Greubel, <a href="/A204924/b204924.txt">Table of n, a(n) for n = 1..1000</a>
%e A204924 1 divides s(2)-s(1), so a(1)=2
%e A204924 2 divides s(3)-s(1), so a(2)=3
%e A204924 3 divides s(4)-s(2), so a(3)=4
%e A204924 9 divides s(7)-s(3), so a(9)=7
%t A204924 s[n_] := s[n] = Fibonacci[n + 1]; z1 = 300; z2 = 60;
%t A204924 Table[s[n], {n, 1, 30}]
%t A204924 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A204924 Table[u[m], {m, 1, z1}]   (* A204922 *)
%t A204924 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A204924 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A204924 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A204924 Table[d[n], {n, 1, z2}]   (* A204923 *)
%t A204924 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A204924 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A204924 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A204924 Table[k[n], {n, 1, z2}]       (* A204924 *)
%t A204924 Table[j[n], {n, 1, z2}]       (* A204925 *)
%t A204924 Table[s[k[n]], {n, 1, z2}]    (* A204926 *)
%t A204924 Table[s[j[n]], {n, 1, z2}]    (* A204927 *)
%t A204924 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204928 *)
%t A204924 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204929 *)
%Y A204924 Cf. A000045, A204892.
%K A204924 nonn
%O A204924 1,1
%A A204924 _Clark Kimberling_, Jan 21 2012