A194085 Strong Achilles numbers: Achilles numbers m such that phi(m) is also an Achilles number, where phi(m) denotes Euler's totient function of m.

500, 864, 1944, 2000, 2592, 3456, 5000, 10125, 10368, 12348, 12500, 16875, 19652, 19773, 30375, 31104, 32000, 33275, 37044, 40500, 49392, 50000, 52488, 55296, 61731, 64827, 67500, 69984, 78608, 80000, 81000, 83349, 84375, 93312, 108000

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..656 from Ray Chandler)

Project Euler, Problem 302: Strong Achilles Numbers

Mathematica

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD @@ ee == 1];
Select[Range[10^6], achillesQ[#] && achillesQ[EulerPhi[#]]&] (* Jean-François Alcover, Sep 26 2020 *)

Prog

(PARI) isA(n)=!ispower(n) && ispowerful(n)
is(n)=isA(n) && isA(eulerphi(n)) \\ Charles R Greathouse IV, Dec 19 2013
(Python)
# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint, totient
def A194085_gen(): # generator of terms
    return map(lambda x:x[0], filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1, map(lambda x: (x, factorint(totient(x)).values()), (A052486(i) for i in count(1)))))
A194085_list = list(islice(A194085_gen(), 20)) # Chai Wah Wu, Sep 10 2024

Crossrefs

Cf. A052486 (Achilles numbers)

Sequence in context: A372147 A372864 A045215 * A216114 A005954 A333137

Adjacent sequences: A194082 A194083 A194084 * A194086 A194087 A194088

Keyword

nonn

Author

Kausthub Gudipati, Aug 14 2011

Extensions

Extended by Ray Chandler, Aug 16 2011

Status

approved