## Friday, 30 December 2011

### Sunday 1 January

Today is 1 day of the year, happy New Year.
1 is the multiplicative identity for the real numbers.
1 is not considered to be a prime number.
1 is not even.
n^0 = 1 for any real n by definition or
$n^{0} = 1, \forall n \epsilon \mathbb{R}$

### Saturday 31 December

Today is 365 and last day of the year.
365 = 5 x 73
365, like 362, is a happy number.
(5^2 + 73^2) / 2 = 2677 which is prime, i.e. the mean of the squares of the prime factors is a prime.
365 = 13^2+14^2 = 10^2+11^2+12^2, i.e. 365 is the sum of both two and three consecutive squares.
365 is palindromic in base 2 (1, 0, 1, 1, 0, 1, 1, 0, 1), base 8 (5, 5, 5), base 14 (1, 12, 1) and base 72 (5, 5).
365 is a member of the following primitive, Pythagorean triples;
(27, 364, 365), (76, 357, 365), (365, 2652, 2677) and (365, 66612, 66613).

### Friday 30 December

Today is 364 day of the year.
364 = 2^2 x 7 x 13
364 is the thirteenth number if the series of tetrahedral (or triangular pyramidal) numbers.
364 is in the following primitive, Pythagorean triples;
(27, 364, 365), (364, 627, 725), (364, 33123, 33125)
364 = 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53, i.e. the sum of twelve consecutive primes.
364 is one of only seven positive numbers where (28 x n) / (28 + n) yields an integer. The complete sequence is 21, 28, 70, 84, 168, 364, 756.
364 is palindromic in base 3 ( 1, 1, 1, 1, 1, 1), base 9 (4, 4, 4), base 25 (14, 14), base 27 (13, 13), base 51 (7, 7), base 90 (4, 4) and base 181 (2, 2).

## Thursday, 29 December 2011

### Thursday 29 December

Today is 363 day of the year.
363 = 3 x 11^2
363 = 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59, i.e. 363 is the sum of nine consecutive primes.
363 is palindromic in base 10 (3, 6, 3), base 32 (11, 11) and base 120 (3, 3).

## Wednesday, 28 December 2011

### Wednesday 28 December

Today is 362 day of the year.
362 = 2 x 181, which makes it semiprime.

The sum of the squares of the digits of 362 is 49,
i.e. (3 x 3) + (6 x 6) + (2 x 2) = 9 + 36 + 4 = 49
Repeating this exercise with the resulting number 49 gives
16 + 81 = 97
Then again with 97 and continuing;
81 + 49 = 130
1 + 9 + 0 = 10
1 + 0 = 1
As the number 1 has been reached then this means that 362 is a happy number.

## Tuesday, 27 December 2011

### Tuesday 27 December

Today is the 361 day of the year.
361 = 19^2, so it is a perfect square, the square of a prime and semiprime.
361 = 105 + 120 + 136 which are the 14th, 15th and 16th triangular numbers.
If one removes the last digit this leaves 36 which is also square.

List all the integers starting at 1 then repeatedly apply the following rules;
Determine the number, n, for this iteration as follows:
If this is the first iteration then the n = 2
otherwise
n is the smallest number remaining in the list that is larger than the previous iterations number.
Delete every nth number in the list.
Thus:
Start with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
The first time iteration has n =2 by definition thus every second number is deleted.
This leaves the odd numbers. 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27
The next iteration removes every third number because 3 is the smallest number in the list that is larger than 2.
This leaves 1, 3, 7, 9, 13, 15, 19, 21, 25, 27
The next iteration removes every seventh number because 7 is the smallest number in the list larger than 3.
This leaves 1, 3, 7, 9, 13, 15, 21, 25, 27
The next iteration removes every ninth number because ....
The resulting sequence is the sequence of lucky numbers. 361 is the 86th number in this sequence.

## Monday, 26 December 2011

### Monday 26 December

Today is the 360 day of the year.
360 = 2^3 x 3^2 x 5
360 = 179 + 181 i.e. two consecutive prime numbers which are twin primes.
360 is the smallest number to have 24 factors which makes it highly composite.
360 is the sum of the 9 consecutive numbers 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44
360 = 91^2 - 89^2 = 23^2 - 13^2 which are the only two ways that it can be expressed as the difference of the squares of two primes. 360 is the second in the sequence of such numbers.

## Sunday, 25 December 2011

### Sunday 25 December

Today is the 359 day of the year. 359 is a prime number.
(359 - 1) / 2 = 179 which is also prime thus 359 is a safe prime.
359 is an Emirp because the reversal of the the digits, 953, is also prime.
3 + 5 + 9 = 17 which is also prime.
Whatever base 359 is written in, from base 2 to base 357, the number is never palindromic:
101100111 (base 2)
111022 (base 3)
11213 (base 4)
2414 (base 5)
1355 (base 6)
1022 (base 7)
547 (base 8)
438 (base 9)
359 (base 10)
2A7 (base 11)
25B ( base 12)
218 (base 13)
1B9 (base 14)
18E (base 15)
167 (base 16)
.
.
.
12 (base 357)

## Saturday, 24 December 2011

### Saturday 24 December

Today is the 358 day of the year.
358 = 2 x 179
By definition this makes 358 a semiprime number.
358 = 47 + 53 + 59 + 61 + 67 + 71, which are six consecutive prime numbers.
The sum of the first 358 prime numbers is 398,771 which is also prime.

## Friday, 23 December 2011

### Friday 23 December 2011

Today is the 357 day of the year.
357 = 3 x 7 x 17
The fact that the prime factorisation consists of three distinct primes, none of which are raised to a power higher than one, means that 357 is a sphenic number by definition. As is shown here, all sphenic numbers have exactly 8 divisors. For 357 these divisors are: 1, 3, 7, 17, 21,  51, 119 and 357.

## Thursday, 22 December 2011

### Thursday 22 December 2011

Today is the 356 day of the year.
356 = 2^2 x 89
Think of a number, say 152.
Add the digits, 1 + 5 + 2 = 8.
Add the two numbers together, 152 + 8 = 160.
There is no number anywhere in the universe that when you apply the three steps above gives an answer of 356. This makes 356 a self number (see http://en.wikipedia.org/wiki/Self_number).
Today is the winter solstice, "the shortest day".
Today is the 40 anniversary of the stretch of the M4 from junction 8/9 to junction 15 being opened.