A343184 - OEIS

%I #41 Mar 01 2022 09:16:51

%S 1,2,127,509,1019,2039,4079,16384,32768,65536,131072,262144,524287,

%T 1048573,6291437

%N Positions of records in A001178.

%C When the Pisano-period map x -> A001175(x) is iterated, a(n) is the number of steps needed to reach a fixed point (that is, to reach a term of A235702).

%C The record values of A001178 at positions 1, 2, 127, 509, 1019, 2039, 4079, 16384, 32768, 65536, 131072, 262144, 524287, 1048573 are 0, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 18, 19.

%C Felix Lans remarks that from a(4) onwards, the terms appear to be very close to consecutive powers of 2, except that there is no term close to 2^13.

%t Block[{pi, k, s}, pi[1] = 1; pi[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k + 1], n] == 1, Return[k]]]; s = Array[Length[FixedPointList[pi, #]] - 2 &, 2^11]; Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]] ] (* _Michael De Vlieger_, May 16 2021, after _Jean-François Alcover_ at A001178  *)

%Y Cf. A001175, A001178, A235702.

%K nonn,more

%O 1,2

%A _N. J. A. Sloane_, May 16 2021, following an email from Felix Lans

%E a(1)-a(12) from Felix Lans and confirmed by _Rémy Sigrist_, May 16 2021; a(13) and a(14) from _Rémy Sigrist_, May 16 2021.

%E a(15) from _Chai Wah Wu_, Mar 01 2022