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

Author

Labos Elemer, Jul 20 2001

Keywords

• nonn
• easy

Examples

			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
		

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.

Crossrefs

Cf. A000005, A000010, A001620, A276044.

Programs

Mathematica

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

PARI

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

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

Extensions

Offset corrected by Jaroslav Krizek, Jul 24 2009

Edited by Jon E. Schoenfield, Nov 13 2016