Does there exist $a_0$, such that $a_n_n=0^infty$ is unbounded?

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Suppose $a_n_n=0^infty$ is a sequence, defined by recurrent relation: $a_n+1 = phi(a_n) + sigma(a_n) - a_n$, where $sigma$ is divisor sum and $phi$ is Euler totient function. Does there exist $a_0$, such that the corresponding $a_n_n=0^infty$ is unbounded?



As $phi(a_n) + sigma(a_n) geq 2a_n$ (see: Is $phi(n) + sigma(n) geq 2n$ always true?), every sequence of this type is monotonously non-decreasing. This means, that it is bounded iff it contains an element $a_n$, such that $phi(a_n) + sigma(a_n) = 2a_n$. We know, that to satisfy this equation, $a_n$ must either be $1$ or prime (see: Find all positive integers $n$ such that $phi(n)+sigma(n)=2n$.). Thus, the question is equivalent to: "Does every such sequence $a_n_n=0^infty$ with $a_0 geq 2$ contain a prime element?". And I do not know how to proceed further.



Any help will be appreciated.




elementary-number-theory prime-numbers recurrence-relations totient-function divisor-sum




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  • You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
    – N. S.
    Aug 24 at 16:28










  • It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
    – Paul
    Aug 29 at 17:56










  • Here the numbers up to 1000
    – Paul
    Aug 29 at 18:32















up vote
15
down vote

favorite
5












Suppose $a_n_n=0^infty$ is a sequence, defined by recurrent relation: $a_n+1 = phi(a_n) + sigma(a_n) - a_n$, where $sigma$ is divisor sum and $phi$ is Euler totient function. Does there exist $a_0$, such that the corresponding $a_n_n=0^infty$ is unbounded?



As $phi(a_n) + sigma(a_n) geq 2a_n$ (see: Is $phi(n) + sigma(n) geq 2n$ always true?), every sequence of this type is monotonously non-decreasing. This means, that it is bounded iff it contains an element $a_n$, such that $phi(a_n) + sigma(a_n) = 2a_n$. We know, that to satisfy this equation, $a_n$ must either be $1$ or prime (see: Find all positive integers $n$ such that $phi(n)+sigma(n)=2n$.). Thus, the question is equivalent to: "Does every such sequence $a_n_n=0^infty$ with $a_0 geq 2$ contain a prime element?". And I do not know how to proceed further.



Any help will be appreciated.




elementary-number-theory prime-numbers recurrence-relations totient-function divisor-sum




share|cite|improve this question




















  • You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
    – N. S.
    Aug 24 at 16:28










  • It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
    – Paul
    Aug 29 at 17:56










  • Here the numbers up to 1000
    – Paul
    Aug 29 at 18:32













up vote
15
down vote

favorite
5









up vote
15
down vote

favorite
5






5





Suppose $a_n_n=0^infty$ is a sequence, defined by recurrent relation: $a_n+1 = phi(a_n) + sigma(a_n) - a_n$, where $sigma$ is divisor sum and $phi$ is Euler totient function. Does there exist $a_0$, such that the corresponding $a_n_n=0^infty$ is unbounded?



As $phi(a_n) + sigma(a_n) geq 2a_n$ (see: Is $phi(n) + sigma(n) geq 2n$ always true?), every sequence of this type is monotonously non-decreasing. This means, that it is bounded iff it contains an element $a_n$, such that $phi(a_n) + sigma(a_n) = 2a_n$. We know, that to satisfy this equation, $a_n$ must either be $1$ or prime (see: Find all positive integers $n$ such that $phi(n)+sigma(n)=2n$.). Thus, the question is equivalent to: "Does every such sequence $a_n_n=0^infty$ with $a_0 geq 2$ contain a prime element?". And I do not know how to proceed further.



Any help will be appreciated.




elementary-number-theory prime-numbers recurrence-relations totient-function divisor-sum




share|cite|improve this question












Suppose $a_n_n=0^infty$ is a sequence, defined by recurrent relation: $a_n+1 = phi(a_n) + sigma(a_n) - a_n$, where $sigma$ is divisor sum and $phi$ is Euler totient function. Does there exist $a_0$, such that the corresponding $a_n_n=0^infty$ is unbounded?



As $phi(a_n) + sigma(a_n) geq 2a_n$ (see: Is $phi(n) + sigma(n) geq 2n$ always true?), every sequence of this type is monotonously non-decreasing. This means, that it is bounded iff it contains an element $a_n$, such that $phi(a_n) + sigma(a_n) = 2a_n$. We know, that to satisfy this equation, $a_n$ must either be $1$ or prime (see: Find all positive integers $n$ such that $phi(n)+sigma(n)=2n$.). Thus, the question is equivalent to: "Does every such sequence $a_n_n=0^infty$ with $a_0 geq 2$ contain a prime element?". And I do not know how to proceed further.



Any help will be appreciated.





elementary-number-theory prime-numbers recurrence-relations totient-function divisor-sum





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asked Aug 21 at 13:39









Yanior Weg

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  • You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
    – N. S.
    Aug 24 at 16:28










  • It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
    – Paul
    Aug 29 at 17:56










  • Here the numbers up to 1000
    – Paul
    Aug 29 at 18:32

















  • You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
    – N. S.
    Aug 24 at 16:28










  • It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
    – Paul
    Aug 29 at 17:56










  • Here the numbers up to 1000
    – Paul
    Aug 29 at 18:32
















You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
– N. S.
Aug 24 at 16:28




You mean "Does there exists an $a_0$ such that the sequence doesn't contain a prime element", don't you?
– N. S.
Aug 24 at 16:28












It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
– Paul
Aug 29 at 17:56




It looks like there are a lot of numbers with this property, with $a_0 = 22$ being the smallest.
– Paul
Aug 29 at 17:56












Here the numbers up to 1000
– Paul
Aug 29 at 18:32





Here the numbers up to 1000
– Paul
Aug 29 at 18:32
















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