A062821 Number of divisors of totient of n.

1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 3, 6, 4, 4, 4, 5, 4, 6, 4, 6, 4, 4, 4, 6, 6, 6, 6, 6, 4, 8, 5, 6, 5, 8, 6, 9, 6, 8, 5, 8, 6, 8, 6, 8, 4, 4, 5, 8, 6, 6, 8, 6, 6, 8, 8, 9, 6, 4, 5, 12, 8, 9, 6, 10, 6, 8, 6, 6, 8, 8, 8, 12, 9, 8, 9, 12, 8, 8, 6, 8, 8, 4, 8, 7, 8, 8, 8, 8, 8, 12, 6, 12, 4, 12, 6, 12, 8, 12, 8

Offset

1,3

Links

Harry J. Smith, Table of n, a(n) for n = 1..2000

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

a(n) = A000005(A000010(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

The number of divisors of phi(n) can be greater than, less than, or equal to the number of divisors of n:
.
  n   phi(n)  d(phi(n))   d(n)
  ==  ======  =========   ====
  10     4        3     <   4
  11    10        4     >   2
  28    12        6     =   6

Mathematica

Array[DivisorSigma[0, EulerPhi[#]]&, 110] (* Harvey P. Dale, Jul 13 2012 *)

Prog

(PARI) a(n) = numdiv(eulerphi(n)); \\ Harry J. Smith, Aug 11 2009

Crossrefs

Cf. A000005, A000010, A001620, A276044.

Sequence in context: A318881 A280338 A339147 * A353862 A338719 A296080

Adjacent sequences: A062818 A062819 A062820 * A062822 A062823 A062824

Keyword

nonn,easy

Author

Labos Elemer, Jul 20 2001

Extensions

Offset corrected by Jaroslav Krizek, Jul 24 2009

Edited by Jon E. Schoenfield, Nov 13 2016

Status

approved