singly even number

singly even number

A

singly even number

is an

even number

divisible

by 2 but by no greater

power of two

. If

n

is a singly even number, it satisfies the

congruence

n

≡

2

mod

4

. The first few

positive

singly even numbers are 2, 6, 10, 14, 18, 22, 26, 30, listed in A016825 of Sloane’s OEIS.

In the binary representation of a positive singly even number, the bit immediately to the left of the least significant bit is 1 (the least significant bit itself is of course 0). Thus a single 1-bit right shift is enough to change the parity to odd. These properties obviously also hold true when representing negative numbers in binary by prefixing the

absolute value

with a minus sign. As it turns out, all this also holds true in two’s complement. Independently of binary representation, we can say that the

p

-adic valuation (

http://planetmath.org/PAdicValuation

) of a singly even number with

p

=

2

is

1

2

.

With only one exception, all singly even numbers are

composite

. In representing a singly even number

n

as

∏

i

=

1

π

⁢

(

n

)

p

i

a

i

,

with

p

i

being the

i

th

prime number

,

a

1

=

1

, all other other

a

i

may have any

nonnegative integer

value.

If

n

is singly even, then the value of

τ

⁢

(

n

)

(the

divisor function

) is even. In fact,

τ

⁢

(

n

)

=

2

⁢

τ

⁢

(

n

2

)

. This is because if the

divisors

of

n

2

are

1

,

d

2

,

d

3

,

…

,

d

τ

⁢

(

n

2

)

-

1

,

n

2

, then the divisors of

n

include all these as well as

2

,

2

⁢

d

2

,

2

⁢

d

3

,

…

,

2

⁢

d

τ

⁢

(

n

2

)

-

1

,

n

. (Singly even numbers therefore have an equal amount of odd divisors as they do even divisors). From this it is easy to deduce the relationship of the values of the

sum of divisors function

σ

⁢

(

x

)

for

n

and

n

2

is

σ

⁢

(

n

)

=

3

⁢

σ

⁢

(

n

2

)

. Because

ϕ

⁢

(

2

)

=

1

(

ϕ

⁢

(

n

)

being Euler’s

totient function

) it is also easy to see that for

n

a singly even number it is the case that

ϕ

⁢

(

n

)

=

ϕ

⁢

(

n

2

)

.

Whereas

(

-

1

)

n

=

1

whether

n

is singly or doubly even, with the

imaginary unit

i

it is the case that

i

n

=

-

1

only when

n

is singly even.

Prepending a 0 to the

sequence

of singly even numbers gives a

continued fraction

related to the

natural log base

e

thus:

e

-

1

e

+

1

=

0

+

1

2

+

1

6

+

1

10

+

1

14

+

…

The

multiplicative encodings

of both Pascal’s triangle and Losanitsch’s triangle consist entirely of singly even numbers.

Singly even numbers also have applications in chemistry. To list just two: the maximum number of electrons in an atomic subshell is a singly even number; the number of polyacenes in a carbon nanotube is also a singly even number.

References

1

I. Lukovits & D. Janezic, “Enumeration of conjugated

circuits

in nanotubes”,

J. Chem. Inf. Comput. Sci.

,

44

(2004): 410 - 414

Title

singly even number

Canonical name

SinglyEvenNumber

Date of creation

2013-03-22 17:42:24

Last modified on

2013-03-22 17:42:24

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

10

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A51

Classification

msc 11A63

Related topic

DoublyEvenNumber

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