A173617 Numbers k such that phi(tau(k)) = rad(k).

1, 4, 8, 32, 36, 192, 288, 768, 972, 1458, 5120, 13122, 326592, 19531250, 22588608, 46137344, 171532242, 110000000000, 132799613957120, 1618481116086272, 506590324238281250, 8358680908399640576, 162805498773679522226642, 198263834416799184651812864, 7852841179377049820122874642432, 4299870835974154129876817272635392

Offset: 1

Author

Michel Lagneau, Feb 22 2010

Keywords

• nonn

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).

Numbers k such that A163109(k) = A007947(k).

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)

Examples

			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.
		

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.

Crossrefs

Cf. A000005 (tau), A000010 (phi), A007947 (rad).

Cf. A001221, A001222, A001414, A005384, A049195, A062069, A163109, A173320.

Programs

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 *)

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

Extensions

a(14)-a(17) from Donovan Johnson, Jul 27 2011

a(18)-a(26) from Amiram Eldar, Feb 08 2025