All Questions
Tagged with prime-factorization sequences-and-series
37
questions
2
votes
1
answer
203
views
Do we know the rate of divergence of the sum of reciprocals of the $k$-almost primes?
A $k$-almost prime is a positive integer having exactly $k$ prime factors, not necessarily distinct. Let $\mathbb{P}_k$ be the set of the $k$-almost primes and let $$ \rho_k(n):=\sum\limits_{\substack{...
- 7,299
0
votes
2
answers
44
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Series of positive factors of a number divided by that number
Let $S_n$ be the sum of the positive factors of $2015^n$, with $n$ being a positive integer approaching infinity. What is $\dfrac{S_n}{2015^n}$?
I might be on the wrong track, but I figure that if $x ...
- 1
3
votes
0
answers
107
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Special $\omega(n)$-sequence
Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$.
The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with
$0\le j\le k-1$. In other words, a ...
- 85.1k
3
votes
0
answers
105
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Do there exist any cycles for these number sequences?
We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$:
$$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$
Where $p_1^{q_1}*\cdots *p_m^{...
- 3,540
9
votes
2
answers
840
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Is 641 the Smallest Factor of any Composite Fermat Number?
Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
- 2,195
1
vote
1
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128
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Is $(1+2+3+…)=(1+2+2^2+2^3+…)(1+3+3^2+…)(1+5+5^2+…)…$?
Are these equal?
$$(1+2+3+…)=(1+2+2^2+…)(1+3+3^2+…)(1+5+5^2+…)…$$
Where the RHS has a series for each prime. Looks like they are the same series by the fundamental theorem of arithmetic.
Every number ...
- 3,115
0
votes
1
answer
1k
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Sum of number of factors of first N numbers [duplicate]
Given a number N ( Value can be large like N < 10^9 ) How can we calculate sum of the number of factors of first N numbers??
Example :
For n = 3
Answer:
= #f(1) + #f(2) + #f(3) --- { #f(n) ->...
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