A203663 Achilles number whose double is also an Achilles number.

432, 972, 1944, 2000, 2700, 3456, 4500, 5292, 5400, 5488, 8748, 9000, 10584, 10800, 12348, 12500, 13068, 15552, 16000, 17496, 18000, 18252, 21168, 21296, 21600, 24300, 24500, 24696, 25000, 26136

Offset: 1

Author

Antonio Roldán, Jan 04 2012

Keywords

• nonn

Comments

Every term is a multiple of 4.

Examples

			15552 is in the sequence because 15552 = 2^6*3^5 (Achilles number) and 15552*2 = 2^7*3^5 is also an Achilles number.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..4148 from Robert Israel)

Crossrefs

Cf. A052486, A143610, A203662.

Programs

Maple

filter:= proc(n) local e2,F;
  e2:= padic:-ordp(n,2);
  if e2 < 2 then return false fi;
  F:= map(t -> t[2], ifactors(n/2^e2)[2]);
  min(F) > 1 and igcd(e2,op(F))=1 and igcd(e2+1,op(F))=1
end proc:
select(filter, [seq(i,i=4..10^5,4)]); # Robert Israel, Jan 16 2018

Mathematica

achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD@@ee == 1];
Select[Range[4, 10^5, 4], achillesQ[#] && achillesQ[2#]&] (* Jean-François Alcover, Sep 25 2020 *)

PARI

achilles(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) } \\ M. F. Hasler, 2010
{ for (n=1, 10^6, if (achilles(n)==1 && achilles(2*n)==1, print1(n,", "))); } \\ Antonio Roldán, Oct 07 2012

Python

# uses program in A052486
from itertools import count, islice
from math import gcd
from sympy import factorint
def A203663_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(x<<1).values()),(A052486(i) for i in count(1)))))
A203663_list = list(islice(A203663_gen(),30)) # Chai Wah Wu, Sep 10 2024