Today is the $50^{th}$ day of the year.
$50$ is an odious number and a Harshad or Niven number.
An odious number is any number that has an odd number of ones in its binary expansion.
$50_{10} = 110010_{2}$ so it has three ones in its binary form and is therefore odious.
The first 31 odious numbers are:
1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 31, 32, 35, 37, 38, 41, 42, 44, 47, 49, 50, 52, 55, 56, 59, 61
Those numbers that are not odious are Evil Numbers.
The first 30 Evil Numbers are:
0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, 23, 24, 27, 29, 30, 33, 34, 36, 39, 40, 43, 45, 46, 48, 51, 53, 54, 57, 58
A Harshad or Niven Number is any number that is divisible by the sum of its digits.
The sum of the digits of $50$ is $5 + 0 = 5$ and, clearly, $50$ is divisible by $5$.
The first 30 Harshad numbers are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81
Tuesday, 19 February 2013
Monday, 18 February 2013
MONDAY, 18 FEBRUARY 2013
Today is $49^{th}$ day of the year.
$49$ is a perfect square.
$49$ is also a lucky number.
The best description of lucky numbers that I have come across is from Ivars Peterson's MathTrak blog which I quote from below:
Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.
To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor. The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.
Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.
Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.
This particular sieving process yields certain numbers that permanently escape getting killed. That's why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.
1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195
These lucky numbers should not be confused with Euler's Lucky Numbers, see http://oeis.org/A014556.
For those that are interested, the $66$ days of this year that are related to a lucky number are:
$49$ is a perfect square.
$49$ is also a lucky number.
The best description of lucky numbers that I have come across is from Ivars Peterson's MathTrak blog which I quote from below:
Hunting for prime numbers, those evenly divisible only by themselves and 1, requires a sieve to separate them from the rest. For example, the sieve of Eratosthenes, named for a Greek mathematician of the third century B.C., generates a list of prime numbers by the process of elimination.
To find all prime numbers less than, say, 100, the hunter writes down all the integers from 2 to 100 in order (1 doesn't count as a prime). First, 2 is circled, and all multiples of 2 (4, 6, 8, and so on) are struck from the list. That eliminates composite numbers that have 2 as a factor. The next unmarked number is 3. That number is circled, and all multiples of 3 are crossed out. The number 4 is already crossed out, and its multiples have also been eliminated. Five is the next unmarked integer. The procedure continues in this way until only prime numbers are left on the list. Though the sieving process is slow and tedious, it can be continued to infinity to identify every prime number.
Other types of sieves isolate different sequences of numbers. Around 1955, the mathematician Stanislaw Ulam (1909-1984) identified a particular sequence made up of what he called "lucky numbers," and mathematicians have been playing with them ever since.
Starting with a list of integers, including 1, the first step is to cross out every second number: 2, 4, 6, 8, and so on, leaving only the odd integers. The second integer not crossed out is 3. Cross out every third number not yet eliminated. This gets rid of 5, 11, 17, 23, and so on. The third surviving number from the left is 7; cross out every seventh integer not yet eliminated: 19, 39, ... Now, the fourth number from the beginning is 9. Cross out every ninth number not yet eliminated, starting with 27.
This particular sieving process yields certain numbers that permanently escape getting killed. That's why Ulam called them "lucky." See the table below for a list of lucky numbers less than 200.
1 3 7 9 13 15 21 25 31 33 37 43 49 51 63 67 69 73 75 79 87 93 99 105 111 115 127 129 133 135 141 151 159 163 169 171 189 193 195
These lucky numbers should not be confused with Euler's Lucky Numbers, see http://oeis.org/A014556.
For those that are interested, the $66$ days of this year that are related to a lucky number are:
| Lucky Number | Date |
| 1 | Tuesday 1 January |
| 3 | Thursday 3 January |
| 7 | Monday 7 January |
| 9 | Wednesday 9 January |
| 13 | Sunday 13 January |
| 15 | Tuesday 15 January |
| 21 | Monday 21 January |
| 25 | Friday 25 January |
| 31 | Thursday 31 January |
| 33 | Saturday 2 February |
| 37 | Wednesday 6 February |
| 43 | Tuesday 12 February |
| 49 | Monday 18 February |
| 51 | Wednesday 20 February |
| 63 | Monday 4 March |
| 67 | Friday 8 March |
| 69 | Sunday 10 March |
| 73 | Thursday 14 March |
| 75 | Saturday 16 March |
| 79 | Wednesday 20 March |
| 87 | Thursday 28 March |
| 93 | Wednesday 3 April |
| 99 | Tuesday 9 April |
| 105 | Monday 15 April |
| 111 | Sunday 21 April |
| 115 | Thursday 25 April |
| 127 | Tuesday 7 May |
| 129 | Thursday 9 May |
| 133 | Monday 13 May |
| 135 | Wednesday 15 May |
| 141 | Tuesday 21 May |
| 151 | Friday 31 May |
| 159 | Saturday 8 June |
| 163 | Wednesday 12 June |
| 169 | Tuesday 18 June |
| 171 | Thursday 20 June |
| 189 | Monday 8 July |
| 193 | Friday 12 July |
| 195 | Sunday 14 July |
| 201 | Saturday 20 July |
| 205 | Wednesday 24 July |
| 211 | Tuesday 30 July |
| 219 | Wednesday 7 August |
| 223 | Sunday 11 August |
| 231 | Monday 19 August |
| 235 | Friday 23 August |
| 237 | Sunday 25 August |
| 241 | Thursday 29 August |
| 259 | Monday 16 September |
| 261 | Wednesday 18 September |
| 267 | Tuesday 24 September |
| 273 | Monday 30 September |
| 283 | Thursday 10 October |
| 285 | Saturday 12 October |
| 289 | Wednesday 16 October |
| 297 | Thursday 24 October |
| 303 | Wednesday 30 October |
| 307 | Sunday 3 November |
| 319 | Friday 15 November |
| 321 | Sunday 17 November |
| 327 | Saturday 23 November |
| 331 | Wednesday 27 November |
| 339 | Thursday 5 December |
| 349 | Sunday 15 December |
| 357 | Monday 23 December |
| 361 | Friday 27 December |
Sunday, 17 February 2013
SUNDAY, 17 FEBRUARY 2013
Today is $48^{th}$ day of the year.
$48$ is the double factorial of $6$, written $6!!$
The value of the double factorial of a number $n$ is defined as follows:
If $n = -1$ or $n = 0$ then $n!! = 1$ otherwise $n!! = n.(n - 2)!!$
Given $n = 6$ then $6!! = 6.4!! = 6.4.2!! = 6.4.2.0!! = 6.4.2.1 = 48$
However, if $n = 5$ then $5!! = 5.3!! = 5.3.1!! = 5.3.1 = 15$
Thus, $6!!.5!! = 6.4.2.1.5.3.1 = 6.5.4.3.2.1 = 6!$
or, equivalently:
$$ n!!.(n - 1)!! = n! $$
$48$ is the double factorial of $6$, written $6!!$
The value of the double factorial of a number $n$ is defined as follows:
If $n = -1$ or $n = 0$ then $n!! = 1$ otherwise $n!! = n.(n - 2)!!$
Given $n = 6$ then $6!! = 6.4!! = 6.4.2!! = 6.4.2.0!! = 6.4.2.1 = 48$
However, if $n = 5$ then $5!! = 5.3!! = 5.3.1!! = 5.3.1 = 15$
Thus, $6!!.5!! = 6.4.2.1.5.3.1 = 6.5.4.3.2.1 = 6!$
or, equivalently:
$$ n!!.(n - 1)!! = n! $$
Saturday, 16 February 2013
SATURDAY, 16 FEBRUARY 2013
Today is $47^{th}$ day of the year.
47 is a prime number.
47 is the $8^{th}$ Lucas Number, see http://oeis.org/A000032
The Lucas Number are like the Fibonacci numbers in that the $n^{th}$ number is the sum of the previous two numbers, i.e. the $(n-1)^{th}$ and the $(n-2)^{th}$. In the case of the Fibonacci numbers the first two numbers are 0 and 1 giving the sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
whereas the Lucas Numbers start with 2 and 1 giving the following sequence:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, ...
You may be able spot an interesting link between these two series; the $n^{th}$ Lucas Number is equal to the sum of the $(n-1)^{th}$ and the $(n+1)^{th}$ Fibonacci Number i.e the eighth Lucas Number, 47 , is equal to the seventh Fibonacci Number, 13, plus the ninth Fibonacci Number, 34.
We could write this as:
$$L_{n}=F_{n-1}+F_{n+1}$$
Using the same nomenclature, Wikipedia informs us that there are a number of other identities:
\begin{align}L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}\end{align}
\begin{align}L_n^2 = 5 F_n^2 + 4 (-1)^n\end{align}
\begin{align}F_{2n} = L_n F_n\end{align}
\begin{align}F_n = {L_{n-1}+L_{n+1} \over 5}\end{align}
Equation 1
With $m=3$ and $n=5$ then $L_8 = L_4.F_5 + L_3.F_4 = 7.5 + 4.3 = 35 + 12 = 47$
Equation 2
With $n = 8$ then $L_8^2 = 5.F_8^2 + 4.(-1)^8 = 5.{21}^2 + 4 = 5.441 + 4 = 2,209 = 47^2$
Equation 3
With $n = 8$ then $F_{16} = L_8.F_8 = 47.21 = 987$
Equation 4
With $n = 7$ then $F_7 = {L_6 + L_8 \over 5} = {18 + 47 \over 5} = {65 \over 5} = 13$
47 is a prime number.
47 is the $8^{th}$ Lucas Number, see http://oeis.org/A000032
The Lucas Number are like the Fibonacci numbers in that the $n^{th}$ number is the sum of the previous two numbers, i.e. the $(n-1)^{th}$ and the $(n-2)^{th}$. In the case of the Fibonacci numbers the first two numbers are 0 and 1 giving the sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
whereas the Lucas Numbers start with 2 and 1 giving the following sequence:
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, ...
You may be able spot an interesting link between these two series; the $n^{th}$ Lucas Number is equal to the sum of the $(n-1)^{th}$ and the $(n+1)^{th}$ Fibonacci Number i.e the eighth Lucas Number, 47 , is equal to the seventh Fibonacci Number, 13, plus the ninth Fibonacci Number, 34.
We could write this as:
$$L_{n}=F_{n-1}+F_{n+1}$$
Using the same nomenclature, Wikipedia informs us that there are a number of other identities:
\begin{align}L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}\end{align}
\begin{align}L_n^2 = 5 F_n^2 + 4 (-1)^n\end{align}
\begin{align}F_{2n} = L_n F_n\end{align}
\begin{align}F_n = {L_{n-1}+L_{n+1} \over 5}\end{align}
Equation 1
With $m=3$ and $n=5$ then $L_8 = L_4.F_5 + L_3.F_4 = 7.5 + 4.3 = 35 + 12 = 47$
Equation 2
With $n = 8$ then $L_8^2 = 5.F_8^2 + 4.(-1)^8 = 5.{21}^2 + 4 = 5.441 + 4 = 2,209 = 47^2$
Equation 3
With $n = 8$ then $F_{16} = L_8.F_8 = 47.21 = 987$
Equation 4
With $n = 7$ then $F_7 = {L_6 + L_8 \over 5} = {18 + 47 \over 5} = {65 \over 5} = 13$
Friday, 15 February 2013
FRIDAY, 15 FEBRUARY 2013
Today is 46th day of the year.
Take one of the pizzas mentioned in yesterdays post and make nine straight cuts with a pizza cutter. What is the maximum number of pieces of pizza (not necessarily the same size) that can be created with this approach? Not surprisingly it 46. An illustration of the first few members of this sequence, the Lazy Caterer's Sequence, can be found at http://oeis.org/A000124/a000124.gif. It looks like this:
Take one of the pizzas mentioned in yesterdays post and make nine straight cuts with a pizza cutter. What is the maximum number of pieces of pizza (not necessarily the same size) that can be created with this approach? Not surprisingly it 46. An illustration of the first few members of this sequence, the Lazy Caterer's Sequence, can be found at http://oeis.org/A000124/a000124.gif. It looks like this:
Thursday, 14 February 2013
THURSDAY, 14 FEBRUARY 2013
Today is the 45th day of the year.
If you had a choice of two pizza toppings from a selection of 10 then there would be 45 different combinations from which to choose.
Let us assume that the available toppings are Pepperoni, Cheese, Sausage, Mushrooms, Pineapple, Bacon, Ham, Shrimp, Onions and Green Peppers, then the daily toppings could have been:
If you had a choice of two pizza toppings from a selection of 10 then there would be 45 different combinations from which to choose.
Let us assume that the available toppings are Pepperoni, Cheese, Sausage, Mushrooms, Pineapple, Bacon, Ham, Shrimp, Onions and Green Peppers, then the daily toppings could have been:
| Date | Topping 1 | Topping 2 |
| Tuesday 01 January | Pepperoni | Cheese |
| Wednesday 02 January | Pepperoni | Sausage |
| Thursday 03 January | Pepperoni | Mushrooms |
| Friday 04 January | Pepperoni | Pineapple |
| Saturday 05 January | Pepperoni | Bacon |
| Sunday 06 January | Pepperoni | Ham |
| Monday 07 January | Pepperoni | Shrimp |
| Tuesday 08 January | Pepperoni | Onions |
| Wednesday 09 January | Pepperoni | Green Peppers |
| Thursday 10 January | Cheese | Sausage |
| Friday 11 January | Cheese | Mushrooms |
| Saturday 12 January | Cheese | Pineapple |
| Sunday 13 January | Cheese | Bacon |
| Monday 14 January | Cheese | Ham |
| Tuesday 15 January | Cheese | Shrimp |
| Wednesday 16 January | Cheese | Onions |
| Thursday 17 January | Cheese | Green Peppers |
| Friday 18 January | Sausage | Mushrooms |
| Saturday 19 January | Sausage | Pineapple |
| Sunday 20 January | Sausage | Bacon |
| Monday 21 January | Sausage | Ham |
| Tuesday 22 January | Sausage | Shrimp |
| Wednesday 23 January | Sausage | Onions |
| Thursday 24 January | Sausage | Green Peppers |
| Friday 25 January | Mushrooms | Pineapple |
| Saturday 26 January | Mushrooms | Bacon |
| Sunday 27 January | Mushrooms | Ham |
| Monday 28 January | Mushrooms | Shrimp |
| Tuesday 29 January | Mushrooms | Onions |
| Wednesday 30 January | Mushrooms | Green Peppers |
| Thursday 31 January | Pineapple | Bacon |
| Friday 01 February | Pineapple | Ham |
| Saturday 02 February | Pineapple | Shrimp |
| Sunday 03 February | Pineapple | Onions |
| Monday 04 February | Pineapple | Green Peppers |
| Tuesday 05 February | Bacon | Ham |
| Wednesday 06 February | Bacon | Shrimp |
| Thursday 07 February | Bacon | Onions |
| Friday 08 February | Bacon | Green Peppers |
| Saturday 09 February | Ham | Shrimp |
| Sunday 10 February | Ham | Onions |
| Monday 11 February | Ham | Green Peppers |
| Tuesday 12 February | Shrimp | Onions |
| Wednesday 13 February | Shrimp | Green Peppers |
| Thursday 14 February | Onions | Green Peppers |
Wednesday, 13 February 2013
WEDNESDAY, 13 FEBRUARY 2013
Today is $44^{th}$ day of the year.
$$44^{16} + 1 = 197,352,587,024,076,973,231,046,657$$
$197,352,587,024,076,973,231,046,657$ is a prime number.
$$44^{16} + 1 = 197,352,587,024,076,973,231,046,657$$
$197,352,587,024,076,973,231,046,657$ is a prime number.
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