singly even number

A singly even numberMathworldPlanetmath is an even numberMathworldPlanetmath divisible by 2 but by no greater power of two. If n is a singly even number, it satisfies the congruenceMathworldPlanetmathPlanetmathPlanetmath n2mod4. 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 valueMathworldPlanetmathPlanetmathPlanetmath 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 ( of a singly even number with p=2 is 12.

With only one exception, all singly even numbers are composite. In representing a singly even number n as


with pi being the ith prime numberMathworldPlanetmath, a1=1, all other other ai may have any nonnegative integer value.

If n is singly even, then the value of τ(n) (the divisor functionDlmfDlmfMathworldPlanetmath) is even. In fact, τ(n)=2τ(n2). This is because if the divisorsMathworldPlanetmathPlanetmath of n2 are 1,d2,d3,,dτ(n2)-1,n2, then the divisors of n include all these as well as 2,2d2,2d3,,2dτ(n2)-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 n2 is σ(n)=3σ(n2). 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(n2).

Whereas (-1)n=1 whether n is singly or doubly even, with the imaginary unitMathworldPlanetmath i it is the case that in=-1 only when n is singly even.

Prepending a 0 to the sequence of singly even numbers gives a continued fractionDlmfMathworldPlanetmath related to the natural log base e thus:


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 circuitsMathworldPlanetmath 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