A073539 Numbers k such that if p is a prime and divides k then p divides phi(k) (but converse is not a condition for the sequence: phi(k) could have prime factors not dividing k).

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

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

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

Crossrefs

Cf. A000005, A000010, A002322, A007947, A055744, A151999.

Sequence in context: A226385 A070003 A325661 * A090779 A166402 A255586

Adjacent sequences: A073536 A073537 A073538 * A073540 A073541 A073542

Keyword

easy,nonn

Author

Benoit Cloitre, Aug 27 2002

Status

approved