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Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).
1

%I #22 Sep 08 2022 08:45:50

%S 2,3,15,18,24,28,30,33,39,50,52,55,80,132,133,152,169,186,187,190,195,

%T 207,215,217,222,230,238,247,261,266,305,319,333,340,352,369,371,414,

%U 481,484,494,496,497,506,516,522,559,574,580,611,644,646,660,671,689

%N Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).

%C Numbers k such that A000005(A000010(k)) = A000010(A008472(k)).

%D 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 Amiram Eldar, <a href="/A173334/b173334.txt">Table of n, a(n) for n = 1..10000</a>

%H P. A. MacMahon, <a href="http://dx.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 (1921), 75-113.

%H W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Monogr. Matemat. 42 (1964) chapter IV

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_totient_function">Euler's totient function</a>

%F {n : A062821(n)= phi(A008472(n))}.

%e For n=15, tau(phi(15)) = tau(8)=4 equals phi(A008472(15))=phi(8) = 4, which adds 15 to the sequence.

%e For n=18, tau(phi(18)) = tau(6) =4 equals phi(A008472(18)) = phi(5) = 4, which adds 18 to the sequence.

%p with(numtheory): for n from 1 to 1800 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = phi(t2) then print (n): else fi : od :

%t Select[Range[2, 700], DivisorSigma[0, EulerPhi[#]] == EulerPhi[Total[FactorInteger[#][[All, 1]]]] &]

%t (* _Jean-François Alcover_, May 19 2011 *)

%o (Magma) [m:m in [2..700]|#Divisors(EulerPhi(m)) eq EulerPhi(&+PrimeDivisors(m))]; // _Marius A. Burtea_, Jul 10 2019

%o (PARI) isok(n) = numdiv(eulerphi(n)) == eulerphi(vecsum(factor(n)[, 1])); \\ _Michel Marcus_, Jul 10 2019

%Y Cf. A000005, A000010, A008472, A062821.

%K nonn

%O 1,1

%A _Michel Lagneau_, Feb 16 2010

%E Removed sopf acronym. Updated references and links - _R. J. Mathar_, Mar 10 2010