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

1,1

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).

Links

Table of n, a(n) for n=1..70.

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000037, A001105, A005843, A016754.

Equals A183300 \ A005408.

Intersection of A005843 and A183300.

-------------------------------------------------------------------------

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

-------------------------------------------------------------------------

| ... even divisors | A183300 | A001105 |

| ... odd divisors | A028983 | A028982 |

-------------------------------------------------------------------------

Sequence in context: A137230 A369896 A283564 * A181794 A199536 A284883

Adjacent sequences: A348002 A348003 A348004 * A348006 A348007 A348008

Keyword

nonn

Author

Bernard Schott, Oct 02 2021

Status

approved