A254411 -id:A254411 - cp's OEIS frontend

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

Max Alekseyev, Jan 30 2015

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

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

Cf. A245970, A254411.

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 *)

{ 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) }

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.

A254429 a(0) = 0; for n >= 0, a(n+1) = 2^a(n) + 1.

Original entry on oeis.org

0, 2, 5, 33, 8589934593

Offset: 0

Max Alekseyev, Jan 30 2015

nonn

Cf. A007013, A254411.

Cf. A000051.

a254429 n = a254429_list !! n
a254429_list = iterate ((+ 1) . (2 ^)) 0
-- Reinhard Zumkeller, Jan 31 2015

[n le 2 select 2*(n-1) else 2^Self(n-1)+1: n in [1..5]]; // Vincenzo Librandi, Feb 01 2015

RecurrenceTable[{a[0]==0, a[n]==2^a[n-1] + 1}, a, {n, 5}] (* Vincenzo Librandi, Feb 01 2015 *)
NestList[2^#+1&,0,4] (* Harvey P. Dale, Mar 26 2023 *)

a(n+1) = A000051(a(n)). - Reinhard Zumkeller, Jan 31 2015

A318989 Limiting value of A318970(k) mod n as k grows.

Original entry on oeis.org

0, 1, 0, 1, 1, 3, 2, 5, 0, 1, 6, 9, 1, 9, 6, 5, 4, 9, 14, 1, 9, 17, 14, 21, 6, 1, 18, 9, 0, 21, 6, 5, 6, 21, 16, 9, 2, 33, 27, 21, 6, 9, 3, 17, 36, 37, 19, 21, 16, 31, 21, 1, 6, 45, 6, 37, 33, 29, 34, 21, 52, 37, 9, 5, 1, 39, 40, 21, 60, 51, 42, 45, 42, 39, 6, 33, 72, 27, 28, 21, 72, 47, 56, 9, 21, 3, 0, 61, 37, 81, 79, 37, 6, 19, 71, 69, 11, 65, 72, 81

Offset: 1

Max Alekseyev, Sep 06 2018

nonn

Is there a prime p in A318971 such that a(p) is nonzero?

Cf. A245970, A254410, A254411, A318970, A318971.

a(n) = A318970(k) mod n holds for any k >= A227944(n). In particular, a(n) = A318970(A227944(n)) mod n.

cp's OEIS Frontend

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

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A254429 a(0) = 0; for n >= 0, a(n+1) = 2^a(n) + 1.

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A318989 Limiting value of A318970(k) mod n as k grows.

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