All Questions
10
questions
3
votes
1
answer
199
views
Smallest "diamond-number" above some power of ten?
Let us call a positive integer $N$ a "diamond-number" if it has the form $p^2q$ with distinct primes $p,q$ with the same number of decimal digits. An example is $N=10^{19}+93815391$. Its ...
- 85.1k
0
votes
0
answers
76
views
Is the following function $f(k)$ surjective?
Let $\omega(n)$ be the number of distinct prime factors of the positive integer $n$.
For a positive integer $k$ , let $s$ be the smallest positive integer such that $\omega(2024^s+k)\ne s$ , in other ...
- 85.1k
26
votes
1
answer
536
views
Prime factor wanted of the huge number $\sum_{j=1}^{10} j!^{j!}$
What is the smallest prime factor of $$\sum_{j=1}^{10} j!^{j!}$$ ?
Trial :
This number has $23\ 804\ 069$ digits , so if it were prime it would be a record prime. I do not think however that this ...
- 85.1k
4
votes
0
answers
144
views
What are the next primes/semiprimes of the form $\frac{(n-1)^n+1}{n^2}$?
This question is inspired by this question
For an odd positive integer $n$ , define $$f(n):=\frac{(n-1)^n+1}{n^2}$$ as in the linkes question.
For which $n$ is this expression prime , for which $n$ ...
- 85.1k
4
votes
1
answer
178
views
Is the "reverse" of the $33$ rd Fermat number composite?
If we write down the digits of the $33$ rd Fermat number $$F_{33}=2^{2^{33}}+1$$ in base $10$ in reverse order , the resulting number should , considering its magnitude , be composite.
But can we ...
- 85.1k
2
votes
1
answer
102
views
Ratio between largest and smallest prime factor
For a Carmichael-number $N$ let $p$ be the smallest and $q$ the largest prime factor. Consider $r:=\frac{q}{p}$. We could call a Carmichael "balanced" , if $r\approx 1$ , say $r<1.1$.
...
- 85.1k
2
votes
0
answers
127
views
Smallest prime factor of $F_{F_{38}}+F_{38}+1$ wanted
This question is related to this one.
Let $F_n$ be the $n$-th fibonacci number and $p(n)$ be the smallest prime factor of $$f(n):=F_{F_n}+F_n+1$$
The values $p(n)$ except for $n=11$ and $n=38$ upto $n=...
- 85.1k
1
vote
1
answer
87
views
Solution of $\sigma(\sigma(m)+m)=2\sigma(m)$ with $\omega(m)>8$?
This question is related to this one.
$\sigma(n)$ is the divisor-sum function (the sum of the positive divisors of $n$) and $\omega(n)$ is the number of distinct prime factors of $n$.
The object is ...
- 85.1k
1
vote
0
answers
119
views
Is $n=2$ the only even solution of $\sigma(\sigma(n)+n)=2\sigma(n)$?
Inspired by this
question.
For positive integer $m$ , let $\sigma(m)$ be the divisor-sum function.
Let $S$ be the set of positive integers $n$ satisfying $$\sigma(\sigma(n)+n)=2\sigma(n)$$ In the ...
- 85.1k
8
votes
3
answers
505
views
How many numbers are there such that its number of decimal digits equals to the number of its distinct prime factors?
Problem
A positive integer is said to be balanced if the number of its decimal digits equals the number of its distinct prime factors. For instance, $15$ is balanced, while $49$ is not. How many ...
- 3,966