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A073539 Numbers k such that if p is a prime dividing k then p divides phi(k).

%I A073539 #45 Aug 27 2025 11:07:33
%S A073539 1,4,8,9,16,18,25,27,32,36,49,50,54,64,72,81,98,100,108,121,125,128,
%T A073539 144,147,162,169,196,200,216,225,242,243,250,256,288,289,294,324,338,
%U A073539 343,361,392,400,432,441,450,484,486,500,507,512,529,576,578,588,605
%N A073539 Numbers k such that if p is a prime dividing k then p divides phi(k).
%C A073539 Converse does not necessarily hold: phi(k) may have prime factors not dividing k.
%C A073539 Numbers k for which phi(k)*lambda(k) == 0 (mod k), where lambda(k) = A002322(k) is the Carmichael function. - _Michel Lagneau_, Nov 18 2012
%C A073539 Pollack and Pomerance call these numbers "phi-abundant numbers". Numbers k such that rad(k) | phi(k), where rad(k) is the squarefree kernel of k (A007947). - _Amiram Eldar_, Jun 02 2020
%C A073539 If p is the largest prime divisor of a term k, then p^2 divides k. - _Max Alekseyev_, Aug 27 2024
%H A073539 Amiram Eldar, <a href="/A073539/b073539.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A073539 Paul Pollack and Carl Pomerance, <a href="http://www.emis.de/journals/INTEGERS/papers/a14self/a14self.Abstract.html">Prime-Perfect Numbers</a>, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
%e A073539 98 = 2*7^2 and phi(98)=2*3*7 so if p divides 98 then p divides phi(98), hence 98 is in the sequence.
%t A073539 Select[Range[700],And@@Divisible[EulerPhi[#],Transpose[FactorInteger[#]] [[1]]]&] (* _Harvey P. Dale_, Nov 02 2011 *)
%o A073539 (Magma) [n: n in [1..620] | IsZero(EulerPhi(n)^NumberOfDivisors(n) mod n)]; // _Bruno Berselli_, Jul 27 2012
%Y A073539 Cf. A090779, A000010, A002322, A007947, A055744, A151999, A280936.
%K A073539 easy,nonn,changed
%O A073539 1,2
%A A073539 _Benoit Cloitre_, Aug 27 2002