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A173327 Numbers k such that tau(phi(k))= sopf(k).

%I A173327 #25 Sep 08 2022 08:45:50
%S A173327 4,45,48,75,160,180,252,294,300,315,336,351,378,396,475,507,560,605,
%T A173327 616,650,833,936,1216,1375,1452,1690,1805,1920,2023,2112,2200,2349,
%U A173327 2496,2736,3211,3520,3648,4095,4160,4256,4332,4389,4464,4477,4508,4620,4693
%N A173327 Numbers k such that tau(phi(k))= sopf(k).
%C A173327 tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).
%D A173327 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 A173327 Amiram Eldar, <a href="/A173327/b173327.txt">Table of n, a(n) for n = 1..10000</a>
%H A173327 A. Bogomolny, <a href="http://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>
%H A173327 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 A173327 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.
%H A173327 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler&#39;s_phi_function">Euler's totient function</a>
%F A173327 Numbers n such that A062821(n)= A008472(n)
%e A173327 4 is in the sequence because phi(4) = 2, tau(2)=2 and sopf(4)=2 ;
%e A173327 45 is in the sequence because phi(45) = 24, tau(24)=8 and sopf(45)=8.
%p A173327 for n from 1 to 150000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = t2 then print (n): else fi : od :
%t A173327 tpsQ[n_]:=DivisorSigma[0,EulerPhi[n]]==Total[Transpose[FactorInteger[n]][[1]]]; Select[Range[5000],tpsQ] (* _Harvey P. Dale_, Apr 10 2013 *)
%o A173327 (PARI) sopf(n)=my(f=factor(n)[1,]);sum(i=1,#f,f[i])
%o A173327 is(n)=numdiv(eulerphi(n))==sopf(n) \\ _Charles R Greathouse IV_, May 20 2013
%o A173327 (Magma) [ m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+PrimeDivisors(m)]; // _Marius A. Burtea_, Jul 10 2019
%Y A173327 See A173326, A062069. Cf. A001414, A008472(sopfr), A001222, A062821.
%K A173327 nonn
%O A173327 1,1
%A A173327 _Michel Lagneau_, Feb 16 2010
%E A173327 Added punctuation to the examples. Corrected and edited by _Michel Lagneau_, Apr 25 2010
%E A173327 Edited by _D. S. McNeil_, Nov 20 2010