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A166374 Numbers whose arithmetic derivative is equal to Euler totient function: n' = phi(n).

%I A166374 #49 Mar 14 2021 20:41:24
%S A166374 2,9,15,625,1225,3993,117649,218491,857375,3788435,4259571,69302975,
%T A166374 136410197,200533921,313742585,603439225,1516358753,2563893625,
%U A166374 3326174929,5655792025,10214476341,25937424601,29677977573,59797108943,283867750439,715167055525
%N A166374 Numbers whose arithmetic derivative is equal to Euler totient function: n' = phi(n).
%C A166374 The sequence is infinite. If n=prod(pi^ei) with each pi prime, then phi(n) = n*prod((pi-1)/pi) and n' = n*sum(ei/pi). Thus every number of the form p^(p-1), where p is prime, is in this sequence. - _Nathaniel Johnston_, Nov 27 2010
%C A166374 If p > q are primes and q does not divide p-1, there is a solution in positive integers of (p-1)*(q-1) = a*p + b*q, and then p^b*q^a is in the sequence. - _Robert Israel_, Aug 21 2014
%H A166374 Giovanni Resta, <a href="/A166374/b166374.txt">Table of n, a(n) for n = 1..31</a> (terms < 10^13)
%p A166374 A003415:= n -> n*add(f[2]/f[1],f=ifactors(n)[2]):
%p A166374 select(numtheory:-phi = A003415, [$0..10^5]); # _Robert Israel_, Aug 21 2014
%t A166374 (*Run the Mathematica program given in A003415 first, to define the function a as the arithmetic derivative.*) Select[Range[0, 10000], EulerPhi[ # ] == a[ # ] &]
%o A166374 (Python)
%o A166374 from sympy import factorint, totient
%o A166374 A166374 = [n for n in range(1,10**6) if sum([int(n*e/p) for p,e in factorint(n).items()]) == totient(n)] # _Chai Wah Wu_, Aug 22 2014, edited by _Antti Karttunen_, Mar 13 2021
%o A166374 (Sage)
%o A166374 A166374_list = lambda n: filter(lambda k: euler_phi(k) == A003415(k), range(n))
%o A166374 A166374_list(10^6) # _Peter Luschny_, Aug 23 2014
%Y A166374 Cf. A000010, A003415, A036878 (p^(p-1)).
%Y A166374 Intersection of A342008 and A342009.
%K A166374 nonn
%O A166374 1,1
%A A166374 _Paolo P. Lava_ and _Giorgio Balzarotti_, Oct 13 2009
%E A166374 Two terms added by _Alonso del Arte_, Oct 20 2009
%E A166374 Offset corrected and a(12)-a(16) from _Donovan Johnson_, Nov 03 2010
%E A166374 a(17)-a(18) from _Donovan Johnson_, May 09 2011
%E A166374 a(19)-a(24) from _Donovan Johnson_, Oct 01 2012
%E A166374 a(25)-a(28) from _Giovanni Resta_, Mar 13 2014
%E A166374 Term a(1)=0 removed and the indices in the above comments decremented by one. - _Antti Karttunen_, Mar 13 2021