A173743 Numbers k such that phi(tau(k)) = tau(rad(k)).

1, 4, 8, 9, 24, 25, 27, 32, 40, 48, 49, 54, 56, 72, 80, 88, 96, 104, 108, 112, 120, 121, 125, 135, 136, 152, 160, 162, 168, 169, 176, 184, 189, 200, 208, 224, 232, 240, 243, 248, 250, 264, 270, 272, 280, 289, 296, 297, 304, 312, 328, 336, 343, 344, 351, 352, 360

Offset: 1

Author

Michel Lagneau, Feb 23 2010

Keywords

• nonn

Comments

rad(n) = A007947(n). tau(n) = A000005(n). phi(n) = A000010(n). tau(rad(n)) = A034444(n).

Examples

			For n=4, phi(tau(4)) = phi(3)=2 equals tau(rad(4)) = tau(2) = 2, so n=4 is in the sequence.
For n=108, phi(tau(108) ) = phi(12) = 4 equals tau(rad(108)) = tau(6) = 4, so n =108 is in the sequence.
		

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, Table of n, a(n) for n = 1..10000
• W. Sierpinski, Number Of Divisors And Their Sum
• Wikipedia, Euler's totient function

Crossrefs

Cf. A000010, A000005, A007947, A034444, A163109.

Programs

Magma

[ k:k in [1..360]| EulerPhi(#Divisors(k)) eq #Divisors(&*PrimeDivisors(k)) ]; // Marius A. Burtea, Jul 09 2019

Maple

with(numtheory): for n from 1 to 500 do :t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)):if phi(tau(n)) = tau(t2) then print (n): else fi:od:

Mathematica

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[360], EulerPhi[ DivisorSigma[0, #] ] == DivisorSigma[0, rad[#]] &] (* Amiram Eldar, Jul 09 2019 *)

Formula

{ n : A163109(n) = A034444(n) }.