A269818 Numbers coprime to the number of their even divisors.

1, 2, 8, 32, 50, 98, 128, 162, 200, 242, 338, 392, 512, 578, 722, 968, 1058, 1352, 1458, 1568, 1682, 1922, 2048, 2312, 2450, 2592, 2738, 2888, 3200, 3362, 3698, 3872, 4232, 4418, 4802, 5408, 5618, 6050, 6728, 6962, 7442, 7688, 8192, 8450, 8978, 9248, 9800, 10082, 10368, 10658, 10952, 11552, 11858, 12482, 12800

Offset: 1

Author

Waldemar Puszkarz, Mar 05 2016

Keywords

• nonn

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.

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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).

Programs

Mathematica

Select[Range@13000, CoprimeQ[#, Length@Select[Divisors[#], EvenQ]]&]

PARI

for(n=1, 13000, gcd(n, if(n%2, 0, numdiv(n/2)))==1&&print1(n, ", "))