A280936 -id:A280936 - cp's OEIS frontend

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

Benoit Cloitre, Aug 27 2002

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

			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.

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.

Cf. A090779, A000010, A002322, A007947, A055744, A151999, A280936.

[n: n in [1..620] | IsZero(EulerPhi(n)^NumberOfDivisors(n) mod n)]; // Bruno Berselli, Jul 27 2012

Select[Range[700],And@@Divisible[EulerPhi[#],Transpose[FactorInteger[#]] [[1]]]&] (* Harvey P. Dale, Nov 02 2011 *)

A363896 Numbers k such that the sum of primes dividing k (with repetition) is equal to Euler's totient function of k.

Original entry on oeis.org

9, 15, 16, 42

Offset: 1

Darío Clavijo, Jun 26 2023

No more terms less than 1.6*10^7.

Subsequence of A257048.

Other sequences requiring a specific relationship between A000010(k) and A001414(k): A173327, A237798, A280936.

Select[Range[2, 1000], EulerPhi[#] == Plus @@ Times @@@ FactorInteger[#] &] (* Amiram Eldar, Jun 27 2023 *)

is(k) = my(f=factor(k)); f[, 1]~*f[, 2] == eulerphi(f); \\ Amiram Eldar, Jun 27 2023

from sympy import factorint,totient
A001414 = lambda k: sum(p*e for p, e in factorint(k).items())
def g():
  k = 2
  while True:
    if A001414(k) == totient(k): yield(k)
    k += 1
for a_n in g():
  print(a_n)

{k : A001414(k) = A000010(k)}.

cp's OEIS Frontend

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

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