OFFSET
1,1
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)
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[ # ] &]
PROG
(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(10^6) # Peter Luschny, Aug 23 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Paolo P. Lava and Giorgio Balzarotti, Oct 13 2009
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
STATUS
approved