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

%I A173743 #14 Sep 08 2022 08:45:50
%S A173743 1,4,8,9,24,25,27,32,40,48,49,54,56,72,80,88,96,104,108,112,120,121,
%T A173743 125,135,136,152,160,162,168,169,176,184,189,200,208,224,232,240,243,
%U A173743 248,250,264,270,272,280,289,296,297,304,312,328,336,343,344,351,352,360
%N A173743 Numbers k such that phi(tau(k)) = tau(rad(k)).
%C A173743 rad(n) = A007947(n). tau(n) = A000005(n). phi(n) = A000010(n). tau(rad(n)) = A034444(n).
%D A173743 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.
%H A173743 Amiram Eldar, <a href="/A173743/b173743.txt">Table of n, a(n) for n = 1..10000</a>
%H A173743 W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>
%H A173743 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler&#39;s_phi_function">Euler's totient function</a>
%F A173743 { n : A163109(n) = A034444(n) }.
%e A173743 For n=4, phi(tau(4)) = phi(3)=2 equals tau(rad(4)) = tau(2) = 2, so n=4 is in the sequence.
%e A173743 For n=108, phi(tau(108) ) = phi(12) = 4 equals tau(rad(108)) = tau(6) = 4, so n =108 is in the sequence.
%p A173743 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:
%t A173743 rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[360], EulerPhi[ DivisorSigma[0, #] ] == DivisorSigma[0, rad[#]] &] (* _Amiram Eldar_, Jul 09 2019 *)
%o A173743 (Magma) [ k:k in [1..360]| EulerPhi(#Divisors(k)) eq #Divisors(&*PrimeDivisors(k)) ]; // _Marius A. Burtea_, Jul 09 2019
%Y A173743 Cf. A000010, A000005, A007947, A034444, A163109.
%K A173743 nonn
%O A173743 1,2
%A A173743 _Michel Lagneau_, Feb 23 2010