singly even number
A singly even number is an even number divisible by 2 but by no greater power of two. If is a singly even number, it satisfies the congruence . 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 -adic valuation (http://planetmath.org/PAdicValuation) of a singly even number with is .
With only one exception, all singly even numbers are composite. In representing a singly even number as
If is singly even, then the value of (the divisor function) is even. In fact, . This is because if the divisors of are , then the divisors of include all these as well as . (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 for and is . Because ( being Euler’s totient function) it is also easy to see that for a singly even number it is the case that .
Whereas whether is singly or doubly even, with the imaginary unit it is the case that only when is singly even.
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.
- 1 I. Lukovits & D. Janezic, “Enumeration of conjugated circuits in nanotubes”, J. Chem. Inf. Comput. Sci., 44 (2004): 410 - 414
|Title||singly even number|
|Date of creation||2013-03-22 17:42:24|
|Last modified on||2013-03-22 17:42:24|
|Last modified by||CompositeFan (12809)|