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A073539
Numbers k such that if p is a prime dividing k then p divides phi(k).
6
1, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 81, 98, 100, 108, 121, 125, 128, 144, 147, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 294, 324, 338, 343, 361, 392, 400, 432, 441, 450, 484, 486, 500, 507, 512, 529, 576, 578, 588, 605
OFFSET
1,2
COMMENTS
Converse does not necessarily hold: phi(k) may have prime factors not dividing k.
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
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
If p is the largest prime divisor of a term k, then p^2 divides k. - Max Alekseyev, Aug 27 2024
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
Paul Pollack and Carl Pomerance, Prime-Perfect Numbers, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 12a, Paper A14, 2012.
EXAMPLE
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.
MATHEMATICA
Select[Range[700], And@@Divisible[EulerPhi[#], Transpose[FactorInteger[#]] [[1]]]&] (* Harvey P. Dale, Nov 02 2011 *)
PROG
(Magma) [n: n in [1..620] | IsZero(EulerPhi(n)^NumberOfDivisors(n) mod n)]; // Bruno Berselli, Jul 27 2012
KEYWORD
easy,nonn
AUTHOR
Benoit Cloitre, Aug 27 2002
STATUS
approved