A343184 Positions of records in A001178.

1, 2, 127, 509, 1019, 2039, 4079, 16384, 32768, 65536, 131072, 262144, 524287, 1048573, 6291437

Offset: 1

Author

N. J. A. Sloane, May 16 2021, following an email from Felix Lans

Keywords

• nonn
• more

Comments

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).

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.

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.

Crossrefs

Cf. A001175, A001178, A235702.

Programs

Mathematica

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  *)

Extensions

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.

a(15) from Chai Wah Wu, Mar 01 2022