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

2, 9, 15, 625, 1225, 3993, 117649, 218491, 857375, 3788435, 4259571, 69302975, 136410197, 200533921, 313742585, 603439225, 1516358753, 2563893625, 3326174929, 5655792025, 10214476341, 25937424601, 29677977573, 59797108943, 283867750439, 715167055525

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Oct 13 2009

Keywords

• nonn

Comments

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

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..31 (terms < 10^13)

Crossrefs

Cf. A000010, A003415, A036878 (p^(p-1)).

Intersection of A342008 and A342009.

Programs

Maple

A003415:= n -> n*add(f[2]/f[1],f=ifactors(n)[2]):
select(numtheory:-phi = A003415, [$0..10^5]); # Robert Israel, Aug 21 2014

Mathematica

(*Run the Mathematica program given in A003415 first, to define the function a as the arithmetic derivative.*) Select[Range[0, 10000], EulerPhi[ # ] == a[ # ] &]

Python

from sympy import factorint, totient
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

Sage

A166374_list = lambda n: filter(lambda k: euler_phi(k) == A003415(k), range(n))
A166374_list(10^6) # Peter Luschny, Aug 23 2014

Extensions

Two terms added by Alonso del Arte, Oct 20 2009

Offset corrected and a(12)-a(16) from Donovan Johnson, Nov 03 2010

a(17)-a(18) from Donovan Johnson, May 09 2011

a(19)-a(24) from Donovan Johnson, Oct 01 2012

a(25)-a(28) from Giovanni Resta, Mar 13 2014

Term a(1)=0 removed and the indices in the above comments decremented by one. - Antti Karttunen, Mar 13 2021