THOUGHTS of the week: July 2013

Friday, 26 July 2013

OEIS Sequence of the Week [July 29, 2013]

A005235: Fortunate numbers: least $m>1$ such that $m+p_n\#$ is a prime number, where $p_n\#$ is the product of the first $n$ prime numbers:

3, 5, 7, 13, 23, 17, 19, 23, 37, 61, 67, 61, 71, 47, 107, 59, 61, 109, 89, 103, 79, 151, 197, 101, 103, 233, 223, 127, 223, 191, 163, 229, 643, 239, 157, 167, 439, 239, 199, 191, 199, 383, 233, 751, 313, 773, 607, 313, 383, 293, 443, 331, 283, 277, 271, 401, 307, 331

Thursday, 25 July 2013

Problem of the Week [July 29, 2013]

For each positive integer $n$ , let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$ . For example, $T(6)=1$ since the only such triangle with perimeter 6 has side lengths 2, 2, and 2. Determine the smallest positive integer $n$ such that $T(n)>2010$ .

Correct answers as well as solutions will be acknowledged.

Interesting Fact of the Week [July 29, 2013]

On December 16 1811, the Mississippi River began to flow backwards due to a powerful earthquake.

Wednesday, 24 July 2013

Equation of the Week [July 29, 2013]

The Fundamental Theorem of Calculus allows one to compute the definite integral of a function using any one of the infinitely many antiderivatives:

$\int_a^b f(x)\:dx=F(b)-F(a)$ ,

where $f$ and $F$ are real-valued functions on the interval $\[a,b]$ , such that for all $x$ in $\[a,b]$ ,

$F'(x)=f(x)$ .

Visual Art Piece of the Week [July 29, 2013]

Pablo Picasso's Three Musicians:

Tuesday, 23 July 2013

Classical Music Selection of the Week [July 29, 2013]

Franz Schubert's Piano Trio in E flat Major op. 100 D 929 2nd movement - Andante con moto:
http://www.youtube.com/watch?v=tKZBmMu06Fg

Quote of the Week [July 29, 2013]

Milord, I should be sorry if I only entertained them. I wished to make them better.

-George Frideric Handel

OEIS Sequence of the Week [July 22, 2013]

A031157: Numbers that are both lucky and prime:

3, 7, 13, 31, 37, 43, 67, 73, 79, 127, 151, 163, 193, 211, 223, 241, 283, 307, 331, 349, 367, 409, 421, 433, 463, 487, 541, 577, 601, 613, 619, 631, 643, 673, 727, 739, 769, 787, 823, 883, 937, 991, 997, 1009, 1021, 1039, 1087, 1093, 1117, 1123

Problem of the Week [July 22, 2013]

Let $ABCD$ be a cyclic quadrilateral with side $AD=d$ , where $d$ is the diameter of the circle. Also, let $AB=a$ , $BC=a$ , and $CD=b$ . If $a$ , $b$ , and $d$ are integers with $a \neq b$ , determine the minimum value of $d$ .

Correct answers as well as solutions will be acknowledged.

Interesting Fact of the Week [July 22, 2013]

The final resting-place for Dr. Eugene Shoemaker - the Moon. The famed U.S. Geological Survey astronomer trained the Apollo astronauts, but never made it into space. Dr. Shoemaker had wanted to be an astronaut but was rejected because of a medical problem. His ashes were placed on board the Lunar Prospector spacecraft before it was launched on January 6, 1998. NASA crashed the probe into a crater on the moon in an attempt to learn if there was water on the moon.

Equation of the Week [July 22, 2013]

Euler's formula states that, for any real number $x$ ,

$e^{ix}=\cos x + i\sin x$ .

A special case of this formula follows when we let $x=\pi$

$e^{i\pi}=-1$ ,

called Euler's identity.

Classical Music Selection of the Week [July 22, 2013]

Johann Sebastien Bach's Brandenburg Concerto no. 2 1st movement - Allegro Moderato:

http://www.youtube.com/watch?v=gNXaJKdr7yk

Quote of the Week [July 22, 2013]

If I have seen further it is by standing on the shoulders of giants.

-Isaac Newton

Visual Art Piece of the Week [July 22, 2013]

Salvador Dali's The Disintegration of the Persistence of Memory:

Monday, 15 July 2013

OEIS Sequence of the Week [July 15, 2013]

A134056: Least positive integer $k$ such that $(n!)!+k$ is prime:

1, 1, 7, 131, 131, 2897, 21997

Problem of the Week [July 15, 2013]

A Skolem sequence of order $n$ is a sequence $\{s_1,s_2,...,s_{2n}\}$ of $2n$ integers satisfying the conditions:

i) For every $k$ in $\{1, 2,3,...,n\}$ , there exist exactly two elements $s_i$ and $s_j$ with $s_i=s_j=k$ .
ii) If $s_i=s_j=k$ with $i<j$ , then $j-i=k$ .

For example, $\{4, 2, 3, 2, 4, 3, 1, 1\}$ is a Skolem sequence of order 4. If $k$ is a non-negative integer, prove that there is no Skolem sequence of order $n$ if $n$ is of the form $4k+2$ or $4k+3$ .

Correct answers as well as solutions will be acknowledged.