Prove that σ(n)≤⌈log2(n)⌉\sigma(n)\le \lceil\log_2(n)\rceil

Let f:N→N be defined as f(1)=1 and if n=∏kr=1pαrr is the prime decomposition of n then: f(n)=k∏r=1(pr−1)αr Let σ:N→N be defined as the function which associates each number with the number of iterations of f it takes until we reach 1, i.e: f(σ(n)−1)(n)≠1f(σ(n))(n)=1 Prove that σ(n)≤⌈log2(n)⌉. I actually have no idea how to solve this, … Read more

Numbers $m = pq^4$ ($p,q$ are distinct primes) for which $m$ divided by the number of its factors is an integer

The $\operatorname{Ionof}$ (Integer on number of factors) of an integer is the integer divided by the number of factors it has. For example, $18$ has $6$ factors so $\operatorname{Ionof}(18) = \frac{18}{6} = 3$, and $27$ has $4$ factors so $\smash{\operatorname{Ionof}(27) = \frac{27}{4} = 6.75}$. This has been asked here — Solving a Word Problem relating … Read more

Name of integers whose prime factorization is exponent free [squarefree]

Let n>1 be a positive integer. Suppose that n is a product of distinct primes n=k∏i=1pαii i.e. αi=1 for every i. Said equivalently: n has no repeated (square) factor. Surely if n=pq we call them semiprimes but beyond that do we have a general term? Would you just say that n is a “product of … Read more

What can we say about the prime factors of ​^{10}10+23​^{10}10+23?

In a video on ultrafinitism I saw a claim that the number ​^{10}10+23 does not have prime factorization. While I don’t accept the premise of ultrafinitism, I got curious, what can we say about the prime factors of this number? ​^{10}10 refers to the hyperoperation tetration. In other words, the number is equal to 10^{10^{10^{10^{10^{10^{10^{10^{10^{10}}}}}}}}}, … Read more

Find exponent of prime $p$ in prime factorization of a number $x$

Say we have a number $x$ such that $$ x = a^{r}.b^{s}.c^{t}.p^{u} $$ Is there a formula or method which can directly give me the exponent of a particular prime in this prime factorization. For small $x$ calculating it is not a problem but when $x$ is of the order of $10^{8}$ finding the exponent … Read more

Find the prime factors of 332−2323^{32}-2^{32}

I’m having a go at BMO 2006/7 Q1 which states: “Find four prime numbers less than 100 which are factors of 332−232.” My working is as follows (basically just follows difference of two squares loads of times): 332−232 =(316+216)(316−216) =(316+216)(38+28)(38−28) =(316+216)(38+28)(34+24)(34−24) =(316+216)(38+28)(34+24)(32+22)(32−22) =(316+216)(38+28)(34+24)(32+22)(3+2)(3−2) =(316+216)(38+28)(34+24)(32+22)(3+2) Now it is simple to get 3 of the primes here: … Read more

Lower Bound of a Factor of M = 2^P – 1, when M is a composite (P is prime).

I was wondering, is there any rule for the smallest factor of M (where M = 2^P – 1, P is a prime) when M is composite. I have an observation, I found the smallest factor for the following P: P Smallest Factor 11 23 23 47 29 233 37 223 59 179951 193 13821503 … Read more

For every natural number nn, f(n)=f(n) = the smallest prime factor of n.n. For example, f(12)=2,f(105)=3f(12) = 2, f(105) = 3

QUESTION: Let f be a continuous function from R to R (where R is the set of all real numbers) that satisfies the following property: For every natural number n, f(n)= the smallest prime factor of n. For example, f(12)=2,f(105)=3. Calculate the following- (a)lim (b) The number of solutions to the equation f(x) = 2016. … Read more

Find the number when sum of its factors is given.

Can we precisely determine the number when the sum of its factors are given? I was solving a problem where the sum of factors of a particular number $n$ was given as $72$ and I had to evaluate $\frac{1}{d_1}+\frac{1}{d_2}+\frac{1}{d_3}…….\frac{1}{d_k}$ where $d_k$ is the factor of the number $n$ and with some hit and trial I … Read more