OFFSET
1,2
COMMENTS
rad(n) is the product of the primes dividing n (A007947), tau(n) is the number of divisors of n (A000005), and phi(n) is Euler totient function (A000010).
a(18) > 10^10. - Donovan Johnson, Jul 27 2011
From Amiram Eldar, Feb 08 2025: (Start)
1 is the only odd term in this sequence.
The number of terms with any given number of divisors is finite.
There are no terms whose number of divisors d equals 2 or in A049195, or when omega(phi(d)) > bigomega(d), where omega = A001221 and bigomega = A001222.
If p is a Sophie Germain prime (A005384), then 2*p^(2*p) is a term. (End)
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
LINKS
Amiram Eldar, Mathematica code for A173617, 2025.
P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.
Wikipedia, Euler's totient function.
EXAMPLE
8 is a term since tau(8) = 4, phi(4) = 2 and rad(8) = 2.
13122 is a term tau(13122) = 18, phi(18) = 6 and rad(13122) = 6.
MAPLE
with(numtheory):for n from 1 to 1000000 do :t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if phi(tau(n)) = t2 then print (n): else fi : od :
MATHEMATICA
(* First program: see the links section. *)
(* Second program: *)
q[k_] := k == 1 || EvenQ[k] && Module[{f = FactorInteger[k]}, EulerPhi[Times @@ (f[[;; , 2]] + 1)] == Times @@ f[[;; , 1]]]; Select[Range[400000], q] (* Amiram Eldar, Feb 08 2025 *)
PROG
(PARI) isok(k) = if(k == 1, 1, if(k % 2, 0, my(f=factor(k)); eulerphi(numdiv(f)) == vecprod(f[, 1]))); \\ Amiram Eldar, Feb 08 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Feb 22 2010
EXTENSIONS
a(14)-a(17) from Donovan Johnson, Jul 27 2011
a(18)-a(26) from Amiram Eldar, Feb 08 2025
STATUS
approved