A268066 Even numbers coprime to the number of their divisors.

4, 16, 64, 100, 196, 256, 484, 676, 784, 1024, 1156, 1296, 1444, 1600, 1936, 2116, 2704, 3364, 3844, 4096, 4624, 4900, 5184, 5476, 5776, 6400, 6724, 7396, 7744, 8464, 8836, 9604, 10816, 11236, 11664, 12100, 12544, 13456, 13924, 14884, 15376, 16384, 16900, 17956, 18496, 20164, 21316, 21904, 23104, 23716, 24964

Offset: 1

Author

Waldemar Puszkarz, Jan 25 2016

Keywords

• nonn

Comments

This is a subsequence of A046642 for even numbers that stands out due to the following property (theorem).

Theorem: even numbers coprime to the number of their divisors are square.

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.

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

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

A046642 (contains this sequence for even terms).

Programs

Maple

select(t -> igcd(t, numtheory:-tau(t))=1, [seq((2*i)^2,i=1..100)]); # Robert Israel, Jan 25 2016

Mathematica

Select[Range[25000], EvenQ[#]&&CoprimeQ[#, DivisorSigma[0,#]]&]
Select[2*Range[13000],CoprimeQ[#,DivisorSigma[0,#]]&] (* Harvey P. Dale, Nov 01 2022 *)

PARI

for(x=1,25000, gcd(x, length(divisors(x)))==1&&(x%2==0)&&print1(x", "))