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

1,3

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.

Formula

Number of divisors of reduced totient function: a(n) = A000005(A002322(n)).

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

Example

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.

Maple

A066800 := proc(n)
    numtheory[tau](numtheory[lambda](n)) ;
end proc:
seq(A066800(n), n=1..100) ; # R. J. Mathar, Oct 01 2017

Mathematica

Array[DivisorSigma[0, CarmichaelLambda@ #] &, 103] (* Michael De Vlieger, Jul 16 2017 *)

Prog

(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

Crossrefs

This is the number of different values of rows of the square array A066799.

Cf. also A206941.

Cf. A000005, A001620, A002322.

Sequence in context: A073668 A302051 A334080 * A368195 A218705 A193459

Adjacent sequences: A066797 A066798 A066799 * A066801 A066802 A066803

Keyword

nonn

Author

Henry Bottomley, Dec 20 2001

Extensions

More terms from David Wasserman, Nov 14 2002

Status

approved