OFFSET
1,2
COMMENTS
This sequence is characterized by the following property (theorem).
Theorem. If n is coprime to the number of its even divisors, then n is 1 or of the form 2m^2, m>0.
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.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MATHEMATICA
Select[Range@13000, CoprimeQ[#, Length@Select[Divisors[#], EvenQ]]&]
PROG
(PARI) for(n=1, 13000, gcd(n, if(n%2, 0, numdiv(n/2)))==1&&print1(n, ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Waldemar Puszkarz, Mar 05 2016
STATUS
approved