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### Counting different residue classes of certain form modulo $4n-1$?

I like to calculate number of different residue classes of the form $b^2c$ modulo $4abc-1$, where $a,b,c$ are positive integers and $\gcd(a,b)=1$. I could prove that if $abc$ is prime there are only 3 ...
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### Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
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### Does 53 diverge to infinity in this Collatz-like sequence?

This function has been explored a bit at MSE (over the past week): \begin{eqnarray} f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\ f(n) &=& \lfloor n/4 \rfloor \; \textrm{...
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### Does anyone recognize this exponential sum?

For $a$, $b$ two integers, let $(a,b)$ denotes their gcd. We define the following exponential sum : $$G_q(n):=\sum_{d|q,~(d,q/d)=1}{e^{2i\pi n\frac{dd'}{q}}}$$ for $n$ a non-negative integer and $q$ ...
Given $n\in\mathbb{N}$, and $f:\mathbb{N}^*\rightarrow \mathbb{N}$, let define $Pos$ as: $$Pos(f)(n)= |\{x \leq n, f(x)=f(n)\}|$$ When given $n\in\mathbb{N}$, this function gives the 'position' of $... 0answers 178 views ### On even almost perfect numbers other than powers of two (Note: This question is an improved version of and has been cross-posted from this MSE post.) Let$\sigma(x)$denote the sum of the divisors of$x$. If$\sigma(x) = 2x - 1$, then we call$xan ... 2answers 403 views ### Asymptotics of product of Euler's totient function (A001088)? Conjecture: \begin{align} \lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...} \end{align} The numerical result from 100000 terms is: My questions ... 2answers 367 views ### Are there multiplicative functions which are not rational? Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ... 1answer 116 views ### Sum of digits of a power [closed] Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number2^{100}$in the system with a base 5. In the system with ... 1answer 227 views ### A question about$(0,1]$-valued multiplicative functions Suppose$f:\mathbb{N}\to [0,1]$is a multiplicative function (i.e.$f(nm)=f(n)f(m)$whenever$m$and$n$are coprime). Suppose$f$has non-zero mean, which means $$\lim_{N\to\infty}\frac{1}{N} \sum_{... 1answer 635 views ### Is the set of multiplicatively even numbers thick? A positive integer is multiplicatively even (odd) if, when decomposed into primes, the sum of the exponents is even (odd). A subset of the integers is thick if it contains arbitrarily long intervals \... 0answers 63 views ### Discrete “difference” equations that involve changes in both shift and scale A standard use of the Z-transform (F(z) = \sum_n (f[n] \cdot z^{-n} )) is to understand the effect of a difference equation on a signal. For instance: y[n] = x[n] + y[n-1] Y(z) = X(z) + Y(z) \... 1answer 137 views ### estimate an sum I need estimate the following sum: \sum_{d=1}^{n}\frac{\mu(d)}{d}\sum_{k=1}^{\lfloor n/d\rfloor}\frac{1}{k}\frac{q^k}{1-q^{-kd}}, where q>1 and \mu is the Möbius function. To obtain the ... 0answers 178 views ### Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula? My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on \mathbb{R}^{+}):$$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$We call ... 1answer 156 views ### Existence of arithmetic function satisfying a certain property I was interested in an arithmetic function satisfying a certain property, I am not sure at the moment if such thing even exists or not. But I was wondering maybe I could get some hint or idea or input ... 1answer 189 views ### sum over primes involving divisor function (variation of the Titchmarsh divisor problem) This question was also asked on MSE. Does there exist an asymptotic estimate for the following sum over primes$$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$where \tau(n)=\sum_{d|n}1 is the divisor ... 0answers 47 views ### All all hypo-multiplicative functions linear combinations of quasi-multiplicative functions? A function is called quasi-multiplicative by many authors if f(m)f(n)=f(1)f(mn), a slight generalization of multiplicativity. (Basically a multiplicative function times a constant is a quasi-... 0answers 232 views ### A closed formula for this arithmetic function The following function comes up in my research as part of a sufficient condition for capability of p-group of class two and prime exponent. Given a nonnegative integer m, express m as a ... 0answers 278 views ### n\varphi(n)\equiv 2\pmod{\sigma(n)} as a primality test It is known from Subbarao, "On two congruences for primality" that n>22 is a prime iff$$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$where \varphi(n) is Euler's function and \sigma(n) is sum of ... 1answer 206 views ### Are all known k-multiperfect numbers (for k > 2) not squarefree? I asked the following question in MSE four (4) days ago, but so far nobody has posted an answer. The gist of the question is as follows: Are all known k-multiperfect numbers (for k > ... 2answers 181 views ### What proportion of the positive integers satisfy I(n^2) < (1 + \frac{1}{n})I(n), where I(x) is the abundancy index of x? Let \sigma(x) denote the classical sum-of-divisors function, and let$$I(x) = \frac{\sigma(x)}{x}$$be the abundancy index of the positive integer x. My question is this: What proportion of ... 1answer 430 views ### Menon’s identity I also put this question in stackexchange, but remained unanswered. http://math.stackexchange.com/questions/506996/menons-identity Let G be a group of order n. Consider an action of U_n, the ... 1answer 259 views ### What proportion of the positive integers satisfy I(n) < \frac{2n}{n + 1} \leq I(n^2) < 2? Let$$I(x) = \frac{\sigma(x)}{x}$$be the abundancy index of the positive integer x. Note that \sigma(x) is the classical sum-of-divisors function. For example,$$\sigma(12) = 1 + 2 + 3 + 4 + ... 0answers 202 views ### Maximal order of Hooley's Delta function? There is a large literature on Hooley's $$\Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1$$ giving its normal and average order. What is known of its maximal order? Clearly$\Delta(n)\le d(n)$... 1answer 641 views ### A formula combining Euler$\phi$and$\gcd$Let us fix a natural number$N>1$and$a_1, \ldots, a_n$natural numbers satisfying$0 \leq a_i < N$, with the property that$1+ \sum a_i$is divisible by$N$. Let$\phi$be the Euler totient ... 5answers 2k views ### The Inverse of the Euler Totient Function How can we calculate the cardinality of the inverse of Totient function of any positive integer n ? I tried going through this paper, but I couldn't understand the procedure. Thanks 3answers 563 views ### The digit sum:$s(na)=s(nb)$Not that I was serious about the following question, but I think it is a must-to-ask as a follow-up to this MO post. For integer$n\ge0$, let$s(n)$denote the sum of the digits in the decimal ... 2answers 1k views ### Sum of$\sum_{k=1}^nd(k^2)$There is a literature dealing with $$\sum_{k\le x}d(f(k))$$ where$f$is an irreducible polynomial and$d(n)$is the number of divisors of$n$. Erdos 1952 shows that the sum$\asymp x\log x,$which ... 0answers 269 views ### Linear combination of multiplicative functions Carlitz showed necessary and sufficient conditions for an arithmetic function to be a linear combination of two multiplicative functions. He mentions the possibility of generalizing to$k$... 2answers 490 views ### Which rationals are sum-of-divisor function quotients Consider the function$\sigma(n)/n$, where$\sigma$is the usual sum-of-divisors function. I read somewhere that it is unknown what rational numbers are in fact values of this function (or at any ... 3answers 639 views ### Extending arithmetic functions to groups Thinking along the lines of Tom Leinster's fascinating recent question, I'm wondering more generally about how to extend questions about natural numbers to groups, with the cyclic groups representing ... 2answers 2k views ### Generalized Euler phi function Let$n$be an integer, there is a well-known formula for$\varphi(n)$where$\varphi$is the Euler phi function. Essentially,$\varphi(n)$gives the number of invertible elements in$\mathbb{Z}/n\...
Original version. I believe that it is an elementary question, already discussed somewhere. But I just have no idea of how to start it properly. Take a positive integer $n=n_1$ and compute its sum of ...