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

1,2

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

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).

Sequence in context: A238984 A071389 A026369 * A331823 A006552 A134753

Adjacent sequences: A269815 A269816 A269817 * A269819 A269820 A269821

Keyword

nonn

Author

Waldemar Puszkarz, Mar 05 2016

Status

approved