The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
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It is a result of Luca that
$$\mathrm{gcd}(C_n,C_m)<\exp(c_2(m \log m)^{1/2})$$
for all but finitely many pairs of positive integers $m>n>c_1$, with $c_1, c_2$ two computable constants.
- Florian Luca, On the greatest common divisor of two Cullen numbers (2003)
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$\begingroup$ And,I have and next question: when Cn divisible by some integer k ? $\endgroup$– daliborCommented Jul 26, 2016 at 20:24
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1$\begingroup$ A comment on an answer is not the best place to pose a new question. $\endgroup$ Commented Jul 26, 2016 at 23:08
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$\begingroup$ For a partial answer to that second question: if $k$ is odd, let the order of $2$ mod $k$ be $m$, a divisor of $\phi(k)$. Then $n 2^n \mod k$ is periodic in $n$ with period $km$. Compute which if any of the first $km$ values $n 2^n \mod k$, and this determines the answer. $\endgroup$ Commented Jul 27, 2016 at 1:19
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$\begingroup$ In particular, if $k$ and $m$ are coprime, this is always possible with $n \equiv -1 \mod k$ and $n \equiv 0 \mod m$. $\endgroup$ Commented Jul 27, 2016 at 1:26