A254410 Limit of f(f(f(...f(2)...))) modulo n as the number of iterations of f(x) = 2^x - 1 grows.

0, 1, 1, 3, 2, 1, 1, 7, 1, 7, 6, 7, 10, 1, 7, 15, 8, 1, 1, 7, 1, 17, 17, 7, 2, 23, 1, 15, 26, 7, 3, 31, 28, 25, 22, 19, 34, 1, 10, 7, 4, 1, 1, 39, 37, 17, 35, 31, 1, 27, 25, 23, 32, 1, 17, 15, 1, 55, 36, 7, 5, 3, 1, 63, 62, 61, 43, 59, 40, 57, 49, 55, 1, 71, 52, 39, 50, 49, 75, 47, 1, 45, 66, 43, 42, 1, 55, 39, 63, 37, 36, 63, 34, 35, 77, 31, 65, 1, 28, 27

Offset: 1

Author

Max Alekseyev, Jan 30 2015

Keywords

• nonn
• easy
• nice
• look

Comments

Also, limit of f(f(f(...f(m)...))) modulo n for any integer m >= 2.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A245970, A254411.

Programs

Mathematica

Clear[a]; Unprotect[Power]; 0^0 = 1; a[1]=0; a[n_] := a[n] = Module[{g, m = n}, g = 2^IntegerExponent[m, 2]; m = Floor[m/g]; Mod[ ChineseRemainder[ {0, Mod[2, m]^a[EulerPhi[m]]}, {g, m}] - 1, n]]; Array[a, 100] (* Jean-François Alcover, Jan 01 2016, adapted from PARI *)

PARI

{ A254410(m) = my(g); if(m==1, return(0)); g=2^valuation(m,2); m\=g; lift( chinese(Mod(0,g),Mod(2,m)^A254410(eulerphi(m)) ) - 1) }

Formula

a(n) = limit of A007013(m) mod n as m grows.

a(n) = A007013(A227944(n) + k) mod n for any k >= 1. In particular, a(n) = A007013(n) mod n.