cp's OEIS Frontend

A269818 Numbers coprime to the number of their even divisors.

%I A269818 #14 Dec 02 2019 04:11:46
%S A269818 1,2,8,32,50,98,128,162,200,242,338,392,512,578,722,968,1058,1352,
%T A269818 1458,1568,1682,1922,2048,2312,2450,2592,2738,2888,3200,3362,3698,
%U A269818 3872,4232,4418,4802,5408,5618,6050,6728,6962,7442,7688,8192,8450,8978,9248,9800,10082,10368,10658,10952,11552,11858,12482,12800
%N A269818 Numbers coprime to the number of their even divisors.
%C A269818 This sequence is characterized by the following property (theorem).
%C A269818 Theorem. If n is coprime to the number of its even divisors, then n is 1 or of the form 2m^2, m>0.
%C A269818 Proof. If n is odd, its number of even divisors is 0 and since gcd(n,0)=|n| (for any n), n must be 1 to be coprime to 0. If n is even, then it is of the form 2^k*p^a*q*^b*...*r^c, where p, q, r are odd primes and k, a, b, c are positive integers, and its sum of even divisors is k*(1+a)*(1+b)*...*(1+c). The latter number can be coprime to an even number only if it is odd, implying that k must be odd and a, b, ..., c must be even; thus n is twice a square.
%H A269818 Amiram Eldar, <a href="/A269818/b269818.txt">Table of n, a(n) for n = 1..10000</a>
%e A269818 For n = 3, a(3) = 8 is a member for the number of even divisors of 8, (2,4,8), is 3, which is coprime with 8.
%t A269818 Select[Range@13000, CoprimeQ[#, Length@Select[Divisors[#], EvenQ]]&]
%o A269818 (PARI) for(n=1, 13000, gcd(n, if(n%2, 0, numdiv(n/2)))==1&&print1(n, ", "))
%Y A269818 Cf. A183063 (number of even divisors), A046642 (numbers coprime to the number of their divisors), A269870 (counterpart for the number of odd divisors), A268066 (related sequence).
%K A269818 nonn
%O A269818 1,2
%A A269818 _Waldemar Puszkarz_, Mar 05 2016