A203663 - OEIS

%I #36 Sep 10 2024 20:50:59

%S 432,972,1944,2000,2700,3456,4500,5292,5400,5488,8748,9000,10584,

%T 10800,12348,12500,13068,15552,16000,17496,18000,18252,21168,21296,

%U 21600,24300,24500,24696,25000,26136

%N Achilles number whose double is also an Achilles number.

%C Every term is a multiple of 4.

%H Chai Wah Wu, <a href="/A203663/b203663.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..4148 from Robert Israel)

%e 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.

%p filter:= proc(n) local e2,F;

%p   e2:= padic:-ordp(n,2);

%p   if e2 < 2 then return false fi;

%p   F:= map(t -> t[2], ifactors(n/2^e2)[2]);

%p   min(F) > 1 and igcd(e2,op(F))=1 and igcd(e2+1,op(F))=1

%p end proc:

%p select(filter, [seq(i,i=4..10^5,4)]); # _Robert Israel_, Jan 16 2018

%t achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD@@ee == 1];

%t Select[Range[4, 10^5, 4], achillesQ[#] && achillesQ[2#]&] (* _Jean-François Alcover_, Sep 25 2020 *)

%o (PARI) achilles(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) } \\ _M. F. Hasler_, 2010

%o { for (n=1, 10^6, if (achilles(n)==1 && achilles(2*n)==1, print1(n,", "))); } \\ _Antonio Roldán_, Oct 07 2012

%o (Python)

%o # uses program in A052486

%o from itertools import count, islice

%o from math import gcd

%o from sympy import factorint

%o def A203663_gen(): # generator of terms

%o     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)))))

%o A203663_list = list(islice(A203663_gen(),30)) # _Chai Wah Wu_, Sep 10 2024

%Y Cf. A052486, A143610, A203662.

%K nonn

%O 1,1

%A _Antonio Roldán_, Jan 04 2012