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A268066 Even numbers coprime to the number of their divisors. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

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

EXAMPLE

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.

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, #]]&]

PROG

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

CROSSREFS

A046642 (contains this sequence for even terms).

Sequence in context: A307138 A114399 A029993 * A275123 A275217 A158988

Adjacent sequences:  A268063 A268064 A268065 * A268067 A268068 A268069

KEYWORD

nonn

AUTHOR

Waldemar Puszkarz, Jan 25 2016

STATUS

approved

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Last modified April 19 09:12 EDT 2021. Contains 343110 sequences. (Running on oeis4.)