All Questions
190
questions
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2
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73
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Expected number of factors of $LCM(1,…,n)$ (particularly, potentially, when $n=8t$)
I’m trying to prove something regarding $8t$-powersmooth numbers (a $k$-powersmooth number $n$ is one for which all prime powers $p^m$ such that $p^m|n$ are such that $p^m\le k$). Essentially, I have ...
- 1,789
3
votes
3
answers
221
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For what integers $n$ does $\varphi(n)=n-5$?
What I have tried so far: $n$ certainly can't be prime. It also can't be a power of prime as $\varphi(p^k)=p^k-p^{k-1})$ unless it is $25=5^2$. From here on, I am pretty stuck. I tried considering the ...
- 637
2
votes
0
answers
57
views
What did I get wrong in this Mobius function question? [closed]
$f(n):=\sum\limits_{d\mid n}\mu(d)\cdot d^2,$ where $\mu(n)$ is the Möbius function. Compute $f(192).$
First, I found all of the divisors of 192 by trial division by primes in ascending order:
$D=\{...
- 637
2
votes
1
answer
97
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Show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs
I am trying to show that $n$ has $2^{\omega(n) - 1}$ coprime factor pairs. I'm pretty sure this is true but I don't see how to prove it. There is no obvious way to use induction.
Here is an example:
$...
- 3,187
2
votes
0
answers
286
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Can factoring $90$ help factor $91$?
There are few posts asking if factoring $N-1$ can help factor $N$. In those posts the focus was on the factors of $N-1$ and $N$ which can never be the same. The conclusion therefore was that ...
- 1,058
3
votes
1
answer
217
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A question about prime factorization of composite Mersenne numbers and $(2^p-2)/(2 \cdot p)$
Mersenne numbers are numbers of the form $2^p-1$ where $p$ is a prime number. Some of them are prime for exemple $2^5-1$ or $2^7-1$ and some of them are composite like $2^{11}-1$ or $2^{23}-1$.
I'm ...
- 141
4
votes
0
answers
236
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Generating a random prime
How can I generate a random prime of the form $2^ab+1$ for small $b$ value without actually creating a list of such primes, and then choose from the list at random?
For example: I can generate a ...
- 737
1
vote
1
answer
79
views
Continued aliquot sums
What happens if one takes the aliquot sum of an integer and then repeats the process so that one takes the aliquot sum of all of those factors that were not reduced to the number 1 on the previous ...
1
vote
1
answer
77
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Legendre's Conjecture and estimating the minimum count of least prime factors in a range of consecutive integers
I recently asked a question on MathOverflow that got me thinking about Legendre's Conjecture.
Consider a range of consecutive integers defined by $R(x+1,x+n) = x+1, x+2, x+3, \dots, x+n$ with $C(x+1,x+...
- 10.5k
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0
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51
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Find upper and/or lower bounds for the least prime $p$ such that $p^n + k$ is the product of $n$ distinct primes
Well, first of all, happy new year to everyone.
I am trying to solve the following problem: "Let $k$ be a fixed natural number. Find the least prime $p$ such that there exists a natural number $...
- 28.7k
3
votes
1
answer
119
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Length of this representation increases really slowly?
$$\def\'{\text{'}}\def\len{\operatorname{len}}$$
A recent Code Golf challenge introduced a "base neutral numbering system". Here I present a slightly modified version, but the idea is the ...
- 562
2
votes
2
answers
290
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A detail from the Fermat-prime-test
This question comes from a review of an older study of properties of the "fermat-primetest", but where I did not get aware of the following detail.
With a small spreadsheet-like program I ...
- 35k
3
votes
2
answers
114
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About the Set of $\mathbb{S}=\{ n | n = a^2+b^2, a, b \in \mathbb{Z}. \}$
About the Set of $\mathbb{S}=\{ n | n = a^2+b^2, a, b \in \mathbb{Z}. \}$
This is also known as OEIS A001481.
I just found an interesting one from this set.
From my favorite identity, Brahmagupta-...
- 2,815
0
votes
0
answers
146
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How do we get the number of prime divisors?
We have a natural square-free number $n$ such that $2^5\cdot 3^6\cdot 5^4\equiv 0 \pmod {\tau(n)}$.
Which is the maximum number of different primes that can divide $n$ ?
$$$$
We have that $\tau(n)$ is ...
- 14k
2
votes
1
answer
94
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Numbers for Testing Integer Factoring Algorithms
I'm looking for a list of numbers with which to test an integer factorization algorithm (for a computer). Something that has numbers harder than the ones I could easily come up with. Do any resources ...
- 41