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A062821 Number of divisors of totient of n.

%I A062821 #29 Aug 18 2024 03:54:51
%S A062821 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,
%T A062821 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,
%U A062821 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
%N A062821 Number of divisors of totient of n.
%H A062821 Harry J. Smith, <a href="/A062821/b062821.txt">Table of n, a(n) for n = 1..2000</a>
%H A062821 Florian Luca and Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/PDF/luca030206.pdf">On the average number of divisors of the Euler function</a>, Publicationes Mathematicae Debrecen, Vol. 70, No. 1-2 (2007), pp. 125-148.
%F A062821 a(n) = A000005(A000010(n)).
%F A062821 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
%e A062821 The number of divisors of phi(n) can be greater than, less than, or equal to the number of divisors of n:
%e A062821 .
%e A062821   n   phi(n)  d(phi(n))   d(n)
%e A062821   ==  ======  =========   ====
%e A062821   10     4        3     <   4
%e A062821   11    10        4     >   2
%e A062821   28    12        6     =   6
%t A062821 Array[DivisorSigma[0,EulerPhi[#]]&,110] (* _Harvey P. Dale_, Jul 13 2012 *)
%o A062821 (PARI) a(n) = numdiv(eulerphi(n)); \\ _Harry J. Smith_, Aug 11 2009
%Y A062821 Cf. A000005, A000010, A001620, A276044.
%K A062821 nonn,easy
%O A062821 1,3
%A A062821 _Labos Elemer_, Jul 20 2001
%E A062821 Offset corrected by _Jaroslav Krizek_, Jul 24 2009
%E A062821 Edited by _Jon E. Schoenfield_, Nov 13 2016