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 A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838. 45
 1, 2, 1, 3, 6, 5, 4, 10, 9, 8, 4, 16, 13, 27, 26, 25, 21, 17, 44, 43, 42, 38, 34, 17, 71, 68, 55, 116, 115, 114, 110, 106, 89, 72, 188, 187, 186, 182, 178, 161, 144, 72, 304, 301, 288, 233, 493, 492, 491, 487, 483, 466, 449, 377, 305, 798, 797, 796, 792, 788 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Let s(n)=F(n+1), where F=A000045 (Fibonacci numbers), so that s=(1,2,3,5,8,13,21,...).  If c is a positive integer, there are infinitely many pairs (k,j) such that c divides s(k)-s(j).  The set of differences s(k)-s(j) is ordered as a sequence at A204922.  Guide to related sequences: c....k..........j..........s(k)-s(j)....[s(k)-s(j)]/c 2....A205837....A205838....A205839......A205840 3....A205842....A205843....A205844......A205845 4....A205847....A205848....A205849......A205850 5....A205852....A205853....A205854......A205855 6....A205857....A205858....A205859......A205860 7....A205862....A205863....A205864......A205865 8....A205867....A205868....A205869......A205870 9....A205872....A205873....A205874......A205875 10...A205877....A205878....A205879......A205880 LINKS EXAMPLE The first six terms match these differences: s(3)-s(1) = 3-1 = 2 = 2*1 s(4)-s(1) = 5-1 = 4 = 2*2 s(4)-s(3) = 5-3 = 2 = 2*1 s(5)-s(2) = 8-2 = 6 = 2*3 s(6)-s(1) = 13-1 = 12 = 2*6 s(6)-s(3) = 13-3 = 10 = 2*5 MATHEMATICA s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60; f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2]; Table[s[n], {n, 1, 30}] u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]] Table[u[m], {m, 1, z1}]   (* A204922 *) 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 = 2; t = d[c]    (* A205556 *) 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}]    (* A205837 *) Table[j[n], {n, 1, z2}]    (* A205838 *) Table[s[k[n]] - s[j[n]], {n, 1, z2}](* A205839 *) Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] * A205840 *) CROSSREFS Cf. A204890, A205587, A205839. Sequence in context: A317788 A120576 A063707 * A171084 A332318 A118287 Adjacent sequences:  A205837 A205838 A205839 * A205841 A205842 A205843 KEYWORD nonn AUTHOR Clark Kimberling, Feb 01 2012 STATUS approved

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Last modified May 28 16:37 EDT 2022. Contains 354119 sequences. (Running on oeis4.)