cp's OEIS Frontend

A203663 Achilles number whose double is also an Achilles number.

%I A203663 #36 Sep 10 2024 20:50:59
%S A203663 432,972,1944,2000,2700,3456,4500,5292,5400,5488,8748,9000,10584,
%T A203663 10800,12348,12500,13068,15552,16000,17496,18000,18252,21168,21296,
%U A203663 21600,24300,24500,24696,25000,26136
%N A203663 Achilles number whose double is also an Achilles number.
%C A203663 Every term is a multiple of 4.
%H A203663 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 A203663 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 A203663 filter:= proc(n) local e2,F;
%p A203663   e2:= padic:-ordp(n,2);
%p A203663   if e2 < 2 then return false fi;
%p A203663   F:= map(t -> t[2], ifactors(n/2^e2)[2]);
%p A203663   min(F) > 1 and igcd(e2,op(F))=1 and igcd(e2+1,op(F))=1
%p A203663 end proc:
%p A203663 select(filter, [seq(i,i=4..10^5,4)]); # _Robert Israel_, Jan 16 2018
%t A203663 achillesQ[n_] := With[{ee = FactorInteger[n][[All, 2]]}, Min[ee] > 1 && GCD@@ee == 1];
%t A203663 Select[Range[4, 10^5, 4], achillesQ[#] && achillesQ[2#]&] (* _Jean-François Alcover_, Sep 25 2020 *)
%o A203663 (PARI) achilles(n) = { n>1 & vecmin(factor(n)[, 2])>1 & !ispower(n) } \\ _M. F. Hasler_, 2010
%o A203663 { for (n=1, 10^6, if (achilles(n)==1 && achilles(2*n)==1, print1(n,", "))); } \\ _Antonio Roldán_, Oct 07 2012
%o A203663 (Python)
%o A203663 # uses program in A052486
%o A203663 from itertools import count, islice
%o A203663 from math import gcd
%o A203663 from sympy import factorint
%o A203663 def A203663_gen(): # generator of terms
%o A203663     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 A203663_list = list(islice(A203663_gen(),30)) # _Chai Wah Wu_, Sep 10 2024
%Y A203663 Cf. A052486, A143610, A203662.
%K A203663 nonn
%O A203663 1,1
%A A203663 _Antonio Roldán_, Jan 04 2012