A203662 Achilles number whose largest proper divisor is also an Achilles number.

864, 1944, 3888, 4000, 5400, 6912, 9000, 10584, 10800, 10976, 17496, 18000, 21168, 21600, 24696, 25000, 26136, 30375, 31104, 32000, 34992, 36000, 36504, 42336, 42592, 43200, 48600, 49000, 49392, 50000, 52272, 55296, 62208, 62424, 68600, 69984

Offset: 1

Author

Antonio Roldán, Jan 04 2012

Keywords

• nonn

Comments

Exponent of smallest prime divisor of n is greater than or equal to 3.

Both N and the largest proper divisor of N share the same prime factors with different exponents.

Examples

			17496 is in the sequence because 17496=2^3*3^7 (Achilles number) and the largest proper divisor 8748=2^2*3^7 is also an Achilles number.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Donovan Johnson)

Crossrefs

Cf. A052486, A143610.

Programs

Mathematica

(* First run the program for A052486 to define achillesQ *) Select[Range[50000], achillesQ[#] && achillesQ[Divisors[#][[-2]]] &] (* Alonso del Arte, Jan 05 2012 *)

Python

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203662_gen(): # generator of terms
    def g(x):
        (f:=factorint(x))[min(f)]-=1
        return (x,f.values())
    return map(lambda x:x[0],filter(lambda x:all(d>1 for d in x[1]) and gcd(*x[1])==1,map(g,(A052486(i) for i in count(1)))))
A203662_list = list(islice(A204662_gen(),20)) # Chai Wah Wu, Sep 10 2024