New Years Day Warm Up.

Here is a warm up for tomorrow.  There are some things you need to know about 2015 before it begins:

The year 2015 will have 13 full moons.

The year 2015 will have 3 Friday the 13^ths.

2015: Time travelers Marty McFly and Doc Brown arrive from the year 1985. (From the movie “Back to the Future Part II”)

2015 has 8 divisors (1, 5, 13, 31, 65, 155, 403, and 2015), whose sum is σ = 2688.  The sum of its proper divisors is 673.  Its totient is φ = 1440.  The sum of its prime factors is 49.  The product of its (nonzero) digits is 10, while the sum is 8.

It is a sphenic number (or 3-almost prime), since it is the product of 3 distinct primes: 2015 = 5 * 13 * 31.

2015 is a composite, deficient, evil, odd and square-free number.

It is a Duffinian number.

It is a zygodrome in base 12.

It is a junction number, because it is equal to n + sod(n) for n = 1993 and 2011.

It is a congruent number.

It is a polite number, since it can be written in 7 ways as a sum of consecutive naturals, for example: 2015 = 50 + 51 + ... + 79 + 80.

It is an arithmetic number, because the mean of its divisors is an integer number (336).

2²⁰¹⁵ is an apocalyptic number.

2015 is a wasteful number, since it uses less digits than its factorization.

2015 can be expressed as the sum of 10 different powers of 2: 2015 = 2⁰ + 2¹ + 2² + 2³ + 2⁴ + 2⁶ + 2⁷ + 2⁸ + 2⁹ + 2¹⁰.

2015 is a trapezoidal number (the difference of two triangular numbers): 2015 equals the 63^rd triangular number minus the 1^st triangular number.

2015 is a 31-smooth and a 5-rough number.

2015 is a palindromic and Cyclops number when written in Base 2: 11111011111₂.

2015 is written MMXV in Roman numerals.

2015 is an undulating number when written in base 8: 3737₈.

2015 can be expressed as the difference of two squares in four different ways: 2015 = 48² - 17² = 84² - 71² = 204² - 199² = 1008² - 1007².

2015 is one of 5 number located between the sexy prime pair of 2011 and 2017.

2015 is a Lucas-Carmichael number.

The string 2015 occurs at position 19038 (counting from the first digit after the decimal point.  The “3.” is not counted.).  This string occurs 20,090 times in the first 200 million digits of Pi.

Have a Happy and Safe New Year.

David