A066800 Number of different eventual period lengths for power sequences mod n; i.e., number of different period lengths of repeating digits of 1/n in different bases.
1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 4, 3, 3, 5, 4, 6, 3, 4, 4, 4, 2, 6, 6, 6, 4, 6, 3, 8, 4, 4, 5, 6, 4, 9, 6, 6, 3, 8, 4, 8, 4, 6, 4, 4, 3, 8, 6, 5, 6, 6, 6, 6, 4, 6, 6, 4, 3, 12, 8, 4, 5, 6, 4, 8, 5, 4, 6, 8, 4, 12, 9, 6, 6, 8, 6, 8, 3, 8, 8, 4, 4, 5, 8, 6, 4, 8, 6, 6, 4, 8, 4, 9, 4, 12, 8, 8, 6, 9, 5, 8
Offset: 1
Keywords
Examples
Modulo 5, powers of 1,6,11 etc. are 1,1,1,1,1,1,...; of 2,7,12 etc. are 1,2,4,3,1,2,4,3,...; of 3,8,13 etc. are 1,3,4,2,1,3,4,2,...; of 4,9,14 etc. are 1,4,1,4,1,4,...; of 5,10,15 etc. are 1,0,0,0,0,... So the eventual period lengths are 1,4,4,2,1 giving three distinct lengths, so a(5)=3.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..10000
- Florian Luca and Carl Pomerance, On the average number of divisors of the Euler function, Publicationes Mathematicae Debrecen, Vol. 70, No. 1-2 (2007), pp. 125-148.
Crossrefs
Programs
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Maple
A066800 := proc(n) numtheory[tau](numtheory[lambda](n)) ; end proc: seq(A066800(n),n=1..100) ; # R. J. Mathar, Oct 01 2017
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Mathematica
Array[DivisorSigma[0, CarmichaelLambda@ #] &, 103] (* Michael De Vlieger, Jul 16 2017 *)
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PARI
A002322(n) = lcm(znstar(n)[2]); \\ This function from Charles R Greathouse IV, Aug 04 2012 A066800(n) = numdiv(A002322(n)); \\ Antti Karttunen, Jul 16 2017
Formula
Sum_{k=1..n} a(k) ~ n * exp(c(n) * (log(n)/log(log(n)))(1/2) * (1 + O(log(log(log(n)))/log(log(n))))), where c(n) is a number in the interval (1/7, 2*sqrt(2))*exp(-gamma/2) and gamma is A001620 (Luca and Pomerance, 2007). - Amiram Eldar, Oct 29 2022
Extensions
More terms from David Wasserman, Nov 14 2002