## Some Interesting Properties of 2015

Happy New Year 2015! Since many Mathematicians are posting fun facts about the number 2015, I thought I would add a few facts myself.

2015 is a composite integer with odd parity, or a parity value of 1. The prime factorisation of 2015 is as follows:

$2015 = \hspace {1mm} 5 \cdot \hspace{1mm} 13 \cdot \hspace{1mm} 31$

2015 is additionally the third Lucas-Carmichael number with 3 prime factors.

There are 8 divisors for 2015, these can be listed as follows:

1, 5, 13, 31, 65, 155, 403 and 2015.

The sum of these divisors is 2688. More formally, we can use the two divisor functions to list the same facts.

$\sigma_0(n) = \sigma_0(2015) = 8$

$\sigma_1(n) = \sigma_1(2015) = 2688$

The binary expansion of 2015 is defined as 11111011111. The binary expansion has three notably interesting facts:

• 2015 is the 51st number whose binary expansion contains a single bit 0.
• 2015 is the 6th number whose binary expansion is palindromic with a single bit 0.
• The binary expansion of 2015 is also evil, since it contains an even number of 1’s.

The square root of 2015 is 44.8888.

$\sqrt{2015} = 44.8888$

2015 is the result the following the equation,

$n \cdot (2n+3)$, where n = 31.

2015/31 = 65 and 2015/65 = 31.

The aliquot sum of 2015 is the following:

$s(n) = \sigma_1(n) - n$, when n is 2015;

$s(2015) = \sigma_1(2015) - 2015 = 673$

2015 is the result of the following quadratic equation,

$8n^2+14n+5$, where n = 15.

2015 is the result of the following equation,

$5n(n+5)/2$, where n = 26.

2015 is the 55nd number, and possibly the largest number (according to OEIS A067528), which satisfies the following condition:

$n-4^k, k > 0 \hspace{1mm} n > 4^k$, with the result being prime or 1.

There are many more interesting Number Theory facts about 2015 on the OEIS database, and through other websites, but the number the properties would easily surpass the size of a reasonable New Year’s post.