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A268066 Even numbers coprime to the number of their divisors.

%I A268066 #12 Nov 01 2022 19:55:56
%S A268066 4,16,64,100,196,256,484,676,784,1024,1156,1296,1444,1600,1936,2116,
%T A268066 2704,3364,3844,4096,4624,4900,5184,5476,5776,6400,6724,7396,7744,
%U A268066 8464,8836,9604,10816,11236,11664,12100,12544,13456,13924,14884,15376,16384,16900,17956,18496,20164,21316,21904,23104,23716,24964
%N A268066 Even numbers coprime to the number of their divisors.
%C A268066 This is a subsequence of A046642 for even numbers that stands out due to the following property (theorem).
%C A268066 Theorem: even numbers coprime to the number of their divisors are square.
%C A268066 Proof: (1) even numbers can be coprime only to odd numbers, (2) a number with an odd number of divisors must be square, (3) to prove (2) let n = p^a*q^b* ... *r^c, where p, q, ..., r are prime and a, b, ..., c positive integers, which gives the number of divisors of n to be (1+a)*(1+b)* ... *(1+c), and if this number is to be odd, all these factors must be odd too, implying a, b, ..., c must be even and thus implying that n must be square.
%C A268066 For n = p_1^e_1 ... p_k^e_k to be a member, where p_j are primes, e_j >= 1 and p_1 = 2, all e_i+1 are coprime to all p_j. - _Robert Israel_, Jan 25 2016
%H A268066 Robert Israel, <a href="/A268066/b268066.txt">Table of n, a(n) for n = 1..10000</a>
%e A268066 For n = 1, a(1) = 4 belongs to this sequence for the number of divisors of 4, (1,2,4), is 3, which makes it coprime with 4.
%p A268066 select(t -> igcd(t, numtheory:-tau(t))=1, [seq((2*i)^2,i=1..100)]); # _Robert Israel_, Jan 25 2016
%t A268066 Select[Range[25000], EvenQ[#]&&CoprimeQ[#, DivisorSigma[0,#]]&]
%t A268066 Select[2*Range[13000],CoprimeQ[#,DivisorSigma[0,#]]&] (* _Harvey P. Dale_, Nov 01 2022 *)
%o A268066 (PARI) for(x=1,25000, gcd(x, length(divisors(x)))==1&&(x%2==0)&&print1(x", "))
%Y A268066 A046642 (contains this sequence for even terms).
%K A268066 nonn
%O A268066 1,1
%A A268066 _Waldemar Puszkarz_, Jan 25 2016