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A173328 Numbers k such that phi(tau(k)) = tau(sopf(k)).

%I A173328 #32 Feb 08 2025 04:52:07
%S A173328 4,6,8,9,10,12,18,20,22,25,27,30,32,34,44,49,50,58,60,68,70,82,90,102,
%T A173328 104,105,116,118,121,125,135,140,142,150,152,164,169,174,182,189,190,
%U A173328 195,202,204,208,214,231,236,238,242,243,246,248,252,274,284,285,286
%N A173328 Numbers k such that phi(tau(k)) = tau(sopf(k)).
%C A173328 Sopf(n) = A008472(n) is the sum of the distinct primes dividing n, tau(n) = A000005(n) is the number of divisors of n, phi = A000010 is Euler's totient function.
%H A173328 Amiram Eldar, <a href="/A173328/b173328.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A173328 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113.
%H A173328 Wacław Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Elementary theory of numbers, Warszawa, 1964.
%F A173328 {k : A163109(k) = tau(A008472(k))}.
%e A173328 4 is in the sequence because tau(4) = 3, phi(3) = 2, sopf(4) = 2 and tau(2) = 2.
%e A173328 6 is in the sequence because tau(6) = 4, phi(6) = 2, sopf(6) = 5 and tau(5) = 2.
%p A173328 isA173328 := proc(n)
%p A173328         numtheory[phi](numtheory[tau](n)) = numtheory[tau](A008472(n)) ;
%p A173328 end proc:
%p A173328 for n from 1 to 300 do
%p A173328         if isA173328(n) then
%p A173328                 printf("%d,",n);
%p A173328         end if;
%p A173328 end do: # _R. J. Mathar_, Nov 07 2011
%t A173328 Select[Range[2,300],EulerPhi[DivisorSigma[0,#]]==DivisorSigma[0, Total[ FactorInteger[#][[All,1]]]]&] (* _Harvey P. Dale_, May 30 2017 *)
%o A173328 (PARI) isok(k) = if(k == 1, 0, my(f=factor(k)); eulerphi(numdiv(f)) == numdiv(vecsum(f[,1]))); \\ _Amiram Eldar_, Feb 08 2025
%Y A173328 Cf. A000005 (tau), A000010 (phi), A008472 (sopfr), A163109.
%K A173328 nonn
%O A173328 1,1
%A A173328 _Michel Lagneau_, Feb 16 2010