2
votes
1answer
43 views

How to find $\sum_{d\mid n}(w(d)w(\frac{n}{d}))$?

i) $w(n)$ is the prime divisor count function. For example $w(6)=2$ ii) Let prime factorization of $n=p_{1}^{a_{1}}p_{2}^{a_{2}}.....p_{w(n)}^{a_{w(n)}}$ iii) Lets define this function. ...
0
votes
2answers
36 views

Different values of $x$ and $y$ between $\sqrt{39}$ and $\sqrt{224}$

If $x$ and $y$ are whole numbers between $\sqrt{39}$ and $\sqrt{224}$, then how many different values can $x$ + $y$ have? OK, first I found that the set numbers are: $$7, 8 ,9 ,10 ,11 ,12, 13,14$$ ...
1
vote
1answer
42 views

question on combinatorics and number theory

We have an equation as: $a\times b < n$ where $n$ is any positive integer. Now my question is how many pairs of positive integers $(a,b)$ can be found to satisfy the equation. For example, ...
0
votes
0answers
57 views

Proportional growing number set $\mathbb{X}\subset\mathbb{N}$

1. Question: Is there such a set of numbers $\mathbb{X}\subset\mathbb{N}$, in which the proportion of product and sum of all natural numbers $n\in \mathbb{N}$ grow proportional? $$\begin{equation*} ...
0
votes
2answers
50 views

A problem about irrational number

I'm dealing with the following problem: By using piegonhole principal, prove that for any positive irrational number $r$ and positive real numbers $x,y \in \left( {0,1} \right)$, $x < y$, there ...
0
votes
1answer
68 views

Count Integers satisfying the conditions

Given some constraints ,I need to find possible ways that these conditions are satisfied. I need to find four POSITIVE integers a,b,c,d such that ad-bc > 0 and also a+d=N for a given value of N. How ...
0
votes
1answer
62 views

What class am I most prepared for?

I've only taken up to calc 3, discrete, and linear algebra. Which course am I most prepared for? I'm going to be taking differential equations and advanced calc, but I want to take a 3rd class. I can ...
1
vote
1answer
18 views

On the number of midpoint free subsets

A set $X$ of real numbers is called midpoint free if whenever $x,y$ are distinct elements of $X$ then $\frac{x+y}2 \not \in X$. What is number of midpoint free subsets of $\{1,2,...,n\}$?
6
votes
1answer
63 views

Gowers' proof of Szemerdi's theorem

Are there any good books or other resources (expository notes) which explains Gowers' proof of Szemerdi's theorem in detail?
1
vote
0answers
58 views

Solving a Recurrence Relation With Summation and Tau Function

How can I solve the following: $$ T(n) = \sum_{i = 1}^{d(n) - 2}T(v_i) + \sum_{i = d(n) - 1}^{n - 1}c + c' $$ Where $d(n)$ is the Tau function, and v is the set of values dividing n. e.g. $d(18) = ...
1
vote
0answers
43 views

How to calculate the number of combinations of different integers under these conditions?

Given a bound, say $1$ - $N$, and $n$ different integers within this bound such that the summation of these $n$ integers is $m$. What is the number of possible combinations of $n$ different integers ...
3
votes
1answer
131 views

Different ways to evaluate $\displaystyle \underbrace{2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}}_n$

Let's take the expression $$\LARGE \underbrace{2^{2^{2^{\cdot^{\cdot^{\cdot^{2}}}}}}}_n$$ and put a nonnegative number of parentheses around them in a logically coherent way. How many possible ...
6
votes
1answer
461 views

Count expressions with 1s and 2s

Moderator's note (20.03.2014) This question is from on on-going contest. Per usual protocol the question will be locked and current answers hidden until the contest ends. Given atmost X number ...
1
vote
3answers
333 views

Divisibility by $8$ for permutations of numbers

Moderator's note: This is an on-going contest problem. Per usual protocol the answers have been hidden and the question is locked until after the contest ends. (21.03.2014) Given an integer $N$. ...
2
votes
0answers
92 views

Please help me remembering a problem on $n!$ [closed]

I couple of years ago a friend of mine gave me a problem about the digits of $n!$, I never solved it and I also forgot what the question was. It asked you to prove some fact about the digits of $n!$ ...
1
vote
1answer
48 views

Solution of a simple linear diophantine equation

I'm having a slight problem with a simple equation of the sort $a_1+a_2+a_3...=n$. Where $n,a_1, a_2, a_3... \in N$. I do know how to find the number of solutions to these equations when they are of ...
0
votes
1answer
40 views

A little question in elementary number theory

How to prove that we can find an increasing sequence, say ${n_k}$ of natural numbers, such that the number of 1's in the binary representation of $3^{n_k}$ is increasing as $k\to \infty$? More ...
4
votes
3answers
150 views

A game with two dice

Imagine a game with two dice, played by two people and a referee. The referee rolls the first die and the number will determine the number of times that the second die will be rolled. The two players ...
0
votes
2answers
48 views

Multiply the money game

Two players A and B are playing a game. The game is as follows: the player having the turn can multiply the money with a particular number between 2 to 9 and pass the money to other player. For ...
1
vote
2answers
80 views

combinatorics and divisibilty

in how many ways we can form a $8$ digit numbers from $1,2,3,4,5$ with repetition allowed & divisible by $8$. MY APPROACH : to be divisible by 8 : last 3 digit of the no. must be divisible by 8 ...
0
votes
1answer
43 views

Number of ways to make grid

I need to construct a L x 3 grid as shown below But i can use only two shapes to make it which are : Here L is the number of small square boxes in each row. I can rotate the shapes as I want. I ...
3
votes
1answer
86 views

Expected Value of this function

Let’s consider a random permutation p1, p2, …, pN of numbers 1, 2, …, N and Function F is calculated as F=(X[2]+…+X[N-1])^K, where ...
0
votes
0answers
35 views

Inclusion-exclusion principle and perfect numbers

I already asked this question on MO but have been told this was not appropriate, so I ask it here. By the way, this question is some kind of a follow-up to my previous question asking whether there is ...
1
vote
0answers
17 views

Combinatorial interpretation of Euclid's form for even perfect numbers

Euclid showed that if $p$ is a prime such that $2^{p}-1$ is also a prime, then the number $n=2^{p-1}.(2^{p}-1)$ is perfect. Much later, Euler proved that every even perfect number is of this form. ...
7
votes
2answers
188 views

Erdős's exercise.

I have tried to solve an exercise I saw in "Topics in the theory of numbers" (Erdős & Suranyi) many times but failed every time I tried. Here it is: Prove that if $a_1,a_2,\cdots$ is an ...
0
votes
2answers
49 views

Number of ways to make particular amount [closed]

How many ways are there to make change for N dollars.There are 5 types of coins in the currency of worth 1 dollar, 5 dollar, 10 dollar,25 dollar and 50 dollar. Example : If N=14 then answer here is ...
3
votes
2answers
100 views

Number-Theoretic Coin Puzzle

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...
0
votes
1answer
40 views

Integer linear combinations of coprime integers

Consider the finite set $S=\{s_1,s_2,\dots,s_n\}$ such that $GCF(s_1,s_2,\dots,s_n)=1$. Show that $\exists n$ such that $n$ cannot be written as $n=c_1s_1+c_2s_2+\dots+c_ns_n \forall c_i,s_i \in ...
1
vote
1answer
22 views

Largest K-multiple free set out of a fully ordered set

i'm struggling conceptually with this problem, i don´t know how to approach it in a clever way (without a computer, or at least without a brilliant algorithm). Mathematicians defined a k-multiple set ...
5
votes
1answer
140 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
1
vote
1answer
32 views

What is the maximum number of iterations before a sequence is repeated

$A = \{a,b,c,d,e\}$ $B = \{f,g,h\}$ $C = \{i,j\}$ $D = \{0,1,2,3,4,5,6\}$ Suppose a four-tuple is constructed by extracting one element from each set at each successive iteration. The stipulation ...
2
votes
2answers
57 views

Describing the pattern in which iterations make two, cyclic sets equal

$A = \{a,b,c,d,e\}$ $B = \{a,b,c\}$ $C = \{0,1,2,3,4,5,6\}$ The first few iterations are as follows: $1.$ $a,a,0$ $2.$ $b,b,1$ $3.$ $c,c,2$ $4.$ $d,a,4$ $5.$ $e,b,5$ $...$ I'm trying to ...
1
vote
2answers
42 views

Game of flipping cards

I have N playing cards on which numbers are written on front as well as back.Now In one move I can flip any card so that its bottom now becomes the top. Given the numbers on top and bottom of cards ...
0
votes
2answers
37 views

Distinct pairs with equal sum mod p

Let $p$ be a prime and $\mathbb{F}$ be a field with $p$ elements. Define the sets $$A=\{ (m_1,m_2) : m_1, m_2 \in \mathbb{F}, m_1 \neq m_2 \}$$ and $$T =\{ (a_{1},a_{2}) : a_1, a_2 \in A, a_1 ...
0
votes
1answer
74 views

Does Bezout's lemma imply you can generate all integers from two co-prime integers?

Does Bezout's lemma imply that: $x\mathbb{Z} + y\mathbb{Z} = \mathbb{Z}$? if $x\in\mathbb{Z}$, $y\in\mathbb{Z}$, $x\neq 0$ and $y\neq0$ and gcd$(x,y)=1$ ($x$ or $y$ can be negative as well)?
1
vote
2answers
55 views

An Identity Involving the Pochhammer Symbol

I need help proving the following identity: $$\frac{(6n)!}{(3n)!} = 1728^n \left(\frac{1}{6}\right)_n \left(\frac{1}{2}\right)_n \left(\frac{5}{6}\right)_n.$$ Here, $$(a)_n = a(a + 1)(a + 2) \cdots (a ...
4
votes
2answers
104 views

How to prove $\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p\;\;$?

An argument that I just came across asserts that if $p$ is prime, and $k, q\in \mathbb{Z}_{>0}$, then $$\frac{1}{q} \binom{p^k q}{p^k} \equiv 1 \mod p \;.$$ This assertion was made in a way (i.e. ...
3
votes
0answers
46 views

News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
2
votes
0answers
30 views

Covering the square with “crosses”.

The problem concerns covering the unit square with translates of a specific figure, which I will refer to as a "cross", using as few translates as possible. The difficulty seems to result from the ...
1
vote
0answers
54 views

Number of times a prime divides a binomial coefficient

Let $E_{p}(n)$ denote the number of times that $p$ divides $n$. (a) show that if $n<p\leq 2n$ then $E_{p}({2n \choose n} )=1.$ (b) Show that if $\dfrac{2}{3}n<p\leq n$ then $E_{p}({2n \choose ...
0
votes
0answers
39 views

Producing integer combinations of irrational numbers in sequence?

Let $\mathbf{w}=\{w_0,w_1,\cdots,w_n\}$, $\mathbf{k}_i=\{k_0^i,k_1^i,\cdots,k_n^i\}$ and $\mathbf{m}_i=\{m_0^i,m_1^i,\cdots,m_n^i\}$, where $w_j\in\mathbb{R}$, $k_j^i\in\mathbb{Z}$ and ...
0
votes
1answer
45 views

How to calculate total number of combination having sum divisible by a given number.

I have following code.And i want to calculate value of ans. ...
7
votes
1answer
108 views

South Africa National Olympiad 2000 (Tile 4xn rectangle using 2x1 tiles)

Let $A_n$ be the number of ways to tile a $4×n$ rectangle using $2×1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3. My attempt: Define basic shapes A, B and C, ...
-1
votes
3answers
77 views

Is there a pair of natural numbers which give the same result for addition and multiplication? [closed]

for example a number is xy0z and another is abcd. + and * them would give same answer. Swaping number positions is allowable. If that is not possible vanish 0 from the numbers. Also tell for other ...
1
vote
0answers
107 views

Number of classes of K-sets

I am having a plane in N dimension. Th distance between 2 points (a1,a2,...,aN) and (b1,b2,...,bN) is max{|a1-b1|, |a2-b2|, ..., |aN-bN|}. I need to to know how many K-sets exist(here K-set refers to ...
2
votes
1answer
70 views

Largest set whose subsets have distinct sums

What is the largest size of a subset $A$ of the integers $\{1,\dots,n\}$ such that no two non-identical subsets of $A$ have the same sum? You can always take the integer powers of $2$ (1,2,4,8, etc.) ...
7
votes
2answers
218 views

Why is Euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime in a clean intuitive way?

Why is does euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime? I had my own proof for it but I really don't like it (it feels not intuitive at all because it ...
1
vote
0answers
53 views

Estimations for the number of prime factors, counted with multiplicity (elementary combinatorics)

If $N_k(x):=\ \mid\{n\leq x : \Omega(n)=k\}\mid$, where $\Omega(n)$ is the number of prime factors (counted with multiplicity) in $n$, I am trying to reason a crude under-estimate for large $k$ and ...
3
votes
2answers
421 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
1
vote
3answers
71 views

Distribute Gifts among students

$N$ students are to be provided with gifts We know that the $i$'th student wants to get at least $a_i$ gifts. The teacher wants to give distinct gifts meaning if he give $x$ gifts to one student then ...