A348005 Positive even integers with an even number of even divisors.

4, 6, 10, 12, 14, 16, 20, 22, 24, 26, 28, 30, 34, 36, 38, 40, 42, 44, 46, 48, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156

Offset: 1

Author

Bernard Schott, Oct 02 2021

Keywords

• nonn

Comments

These terms are exactly the even numbers in A183300.

Complement of A001105 relative to the positive even integers (A005843 \ {0}).

Note that odd integers with an odd number of odd divisors are the odd squares (A016754).

Examples

			The divisors of 14 are {1, 2, 7, 14}, two of them: 2 and 14 are even, hence 14 is a term.
The divisors of 16 are {1, 2, 4, 8, 16}, four of them: 2, 4, 8 and 16 are even, hence 16 is another term.
		

Crossrefs

Cf. A000037, A001105, A005843, A016754.

Equals A183300 \ A005408.

Intersection of A005843 and A183300.

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| Integers with | an even number of ... | an odd number of ... |

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| ... even divisors | A183300 | A001105 |

| ... odd divisors | A028983 | A028982 |

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Programs

Maple

filter:= q -> irem(q, 2) = 0 and sqrt(q/2) <> floor(sqrt(q/2)) : select(filter, [$1..156]);

Mathematica

m = 9; 2 * Complement[Range[m^2], Range[m]^2] (* Amiram Eldar, Oct 02 2021 *)

PARI

isok(k) = !(k % 2) && !(sumdiv(k, d, !(d % 2)) % 2); \\ Michel Marcus, Oct 05 2021

Python

from math import isqrt
def A348005(n): return n+(m:=isqrt(n))+int(n-m*(m+1)>=1)<<1 # Chai Wah Wu, Aug 04 2022

Formula

a(n) = 2*A000037(n).