A173927 -id:A173927 - cp's OEIS frontend

A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

2, 3, 5, 11, 23, 47, 283, 719, 1439, 2879, 34549, 138197, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 1

R. K. Guy

This sequence shares many terms with A056637, the least prime of class n-. Note that 3^(n-1) is an upper bound for each term and the upper bound is reached for n=13 and n=14. Are all subsequent terms 3^(n-1)? The Mathematica code uses the TowerMod function in the CNT package, which is described in the book by Bressoud and Wagon. - T. D. Noe, Mar 13 2009
For n=15, n=16, and n=17, the terms are also of the form 3^(n-1), but for n=18 and n=19, the terms are prime. - Wayne VanWeerthuizen, Aug 26 2014
A185816(a(n)) = n. - Reinhard Zumkeller, Sep 02 2014
Prime terms seen up to n=20 are in eleven instances of the form j*a(n-1)+1, for j=2, 4, 6, or 12. Note, though, that a(2)=5 and a(8)=719 are exceptions to this pattern. - Wayne VanWeerthuizen, Sep 06 2014

			2^^2=2^2=4 and 2^^3=2^2^2=16. We find 4 = 16 (mod k) until k=5. So a(3)=5. - _T. D. Noe_, Mar 13 2009

David Bressoud and Stan Wagon, A Course in Computational Number Theory, Key College Pub., 2000, p. 96.
Stan Wagon, posting to Problem of the Week mailing list, Dec 15 1997.

D. Bressoud, CNT.m Computational Number Theory Mathematica package.
Stan Wagon, Putnam Problem Notes
Eric W. Weisstein, MathWorld: Power Tower

Cf. A056637, A173927, A185816, A246491.

Needs["CNT`"]; k=1; Table[While[TowerMod[2,n,k]==TowerMod[2,n-1,k], k++ ]; k, {n,10}] (* T. D. Noe, Mar 13 2009 *)

Improved the name and changed the offset because I just prepended a term. - T. D. Noe, Mar 13 2009
Corrected and extended by T. D. Noe, Mar 13 2009
Terms a(15)-a(19) from Wayne VanWeerthuizen, Aug 26 2014
Terms a(20)-a(21) from Wayne VanWeerthuizen, Sep 06 2014

A185816 Number of iterations of lambda(n) needed to reach 1.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 2, 3, 3, 3, 3, 4, 3, 4, 3, 3, 4, 5, 2, 4, 3, 4, 3, 4, 3, 4, 3, 4, 4, 3, 3, 4, 4, 3, 3, 4, 3, 4, 4, 3, 5, 6, 3, 4, 4, 4, 3, 4, 4, 4, 3, 4, 4, 5, 3, 4, 4, 3, 4, 3, 4, 5, 4, 5, 3, 4, 3, 4, 4, 4, 4, 4, 3, 4, 3, 5, 4, 5, 3, 4, 4, 4

Offset: 1

Michel Lagneau, Feb 05 2011

nonn

lambda(n) is the Carmichael lambda function, A002322.

a(n) = (length of row n in table A246700) - 1. - Reinhard Zumkeller, Sep 02 2014

			If n = 23 the trajectory is 23, 22, 10, 4, 2, 1. Its length is 6, thus a(23) = 5.

T. D. Noe, Table of n, a(n) for n = 1..10000
Paul Erdős, Andrew Granville, Carl Pomerance, and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204.
Paul Erdős, Andrew Granville, Carl Pomerance, and Claudia Spiro, On the normal behavior of the iterates of some arithmetic functions, Analytic number theory, Birkhäuser Boston, 1990, pp. 165-204. [Annotated copy with A-numbers]
Paul Erdős, A. Granville, C. Pomerance, and C. Spiro, On the Normal Behavior of the Iterates of some Arithmetic Functions, in Analytic number theory (Allerton Park, IL, 1989), Progr. Math., 85 Birkhäuser Boston, Boston, MA, (1990), 165-204.
Nick Harland, The iterated Carmichael lambda function, arXiv:1111.3667v1 [math.NT], Nov 15, 2011.
Nick Harland, The number of iterates of the Carmichael lambda function required to reach 1, arXiv:1203.4791 [math.NT], Mar 21, 2012.

Cf. A002322, A036459, A003434.

Cf. A027763, A173927, A246700.

a185816 n = if n == 1 then 0 else a185816 (a002322 n) + 1
-- Reinhard Zumkeller, Sep 02 2014

a:= n-> `if`(n=1, 0, 1+a(numtheory[lambda](n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Apr 27 2019

f[n_] := Length[ NestWhileList[ CarmichaelLambda, n, Unequal, 2]] - 2; Table[f[n], {n, 1, 120}]

For n > 1: a(n) = a(A002322(n)) + 1. - Reinhard Zumkeller, Sep 02 2014

A256758 Position of first appearance of n in A256757.

Original entry on oeis.org

1, 2, 3, 7, 19, 47, 163, 487, 1307, 2879, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 86093443, 344373773, 688747547, 3486784401

Offset: 0

Ivan Neretin, Apr 09 2015

Smallest number m such that the trajectory of m under iteration of A007733 takes n steps to reach the fixed point.

The terms a(1)..a(9) are primes. The next eight terms are powers of 3, so that for n=10..17, a(n)=3^(n-1), but this apparently established pattern breaks at a(18), which is again a prime.

Cf. A007733, A007755 (similarly built upon the totient function), A173927 (similarly built upon the Carmichael lambda function), A256757.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a256758 = (+ 1) . fromJust . (`elemIndex`  a256757_list)
-- Reinhard Zumkeller, Apr 13 2015

A007733 = Function[n, MultiplicativeOrder[2, n/(2^IntegerExponent[n, 2])]];
A256757 = Function[n, k = 0; m = n; While[m > 1, m = A007733[m]; k++]; k];
a = Function[n, t = 1; While[A256757[t] < n , t++]; t]; Table[a[n], {n, 0, 9}] (* Ivan Neretin, Apr 13 2015 *)

a007733(n) = znorder(Mod(2, n/2^valuation(n, 2)));
a256757(n) = {if (n==1, return(0)); nb = 1; while((n = a007733(n)) != 1, nb++); nb; }
a(n) = {k = 1; while(a256757(k) != n, k++); k;} \\ Michel Marcus, Apr 11 2015

a(15)-a(18) from Michel Marcus, Apr 11 2015

a(19)-a(21) from Amiram Eldar, Mar 04 2023

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A027763 Smallest k such that 2^^n is not congruent to 2^^(n-1) mod k, where 2^^n denotes the power tower 2^2^...^2 (in which 2 appears n times).

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A185816 Number of iterations of lambda(n) needed to reach 1.

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A256758 Position of first appearance of n in A256757.

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