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(Apologies in advance if the terminology is wrong).

I've been led by my research into looking at sequences of primes of the form $(p_1,p_2,\ldots,p_m)$ with each $p_i$ of the form $k_i(p_1-1)+1$, where $k_i$ is a strictly increasing finite sequence of integers. Additionally, the sequences I'm interested in have the property that each $k_i$ divides $k_m$.

For example $(7,13,19,37)$ has this property, with $p_1=7$ and the $k_i = \{1,2,3,6\}$.

$(7,13,19,31)$ does not meet the requirements, because the $k_i$ are $ = {1,2,3,5}$.

Sequences that meet both requirements have the property that $lcm(p_1-1,\ldots,p_m-1)=\lambda(\prod_{i=1}^m p_i)$ (Carmichael's lambda function), is $p_m-1$, the minimum possible value for $m$ primes. Lambda in the first case is $\lambda(7.13.19.37)= 36$, but lambda in the second case is $\lambda(7.13.19.31) = 180$.

Anyone know anything about what happens to the prevalence of such sequences (beyond the fact they become less frequent) as m increases with fixed p1? As p1 increases with fixed m? As the maximum k increases? Any informed speculation? Starting points besides the Prime Number Theorem? Grateful for any help.

reuns
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Barry Fagin
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    There's a lot that can be said (at least speculatively). Can you ask a more specific/quantitiative question than "what happens to the prevalence of such sequences"? – Greg Martin Jan 29 '20 at 01:22
  • I'm wondering if there is something to say given we have the generating function $\sum_n \varphi(n)^{-s}=\prod_p (1+\frac{(p-1)^{-s}}{1-p^{-s}})$ whose logarithm is more or less $\sum_p (p-1)^{-s}$ @GregMartin – reuns Jan 29 '20 at 03:11

2 Answers2

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Given a prime $p_1$, the obvious algorithm is to use Dirichlet's theorem in arithmetic progressions to find $m-2$ primes $p_2,\ldots,p_{m-1}\equiv 1\bmod p_1-1$, then compute $\ell = \lambda(\prod_{i=1}^{m-1} p_i)$ and use Dirichlet's theorem again to find a prime $p_m\equiv 1 \bmod \ell$.

The minimal size of $\lambda(\prod_{i=1}^m p_i)$ as $p_1\to \infty$ depends on the least prime in arithmetic progression as well as the expected size of $\gcd(p_2-1,\ldots,p_{m-1}-1)$. The random model for the primes should predict those quantities reasonably well.

reuns
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  • Greg, not really, I'm not a mathematician, let alone a number theorist, so I'm not really sure what to ask beyond that. Maybe something like are there an infinite number of such sequences if you hold two of the three variables (p1,m,km) fixed? Does the number asymptote to zero?

    Reuns, thanks, please see above for disclaimer on why I miss "obvious" algorithms. Will study Dirichlet's Theorem and the Prime Number Theorem in more detail and see where that gets me. Thanks!

     – Barry Fagin Jan 29 '20 at 01:40
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let $p_1=2j+1$ then we have $q=k_m(2j)+1$, which has 3 divide it; any time the two variables multiply to 1 mod 3, 5 divide it; any time they multiply to 2 mod 5; etc.

One thing of note, is that if $k_m$ are all highly composite or factors of them ( or primorials), you get a related basis to Euclids arguments for an infinite number of primes.

Lastly, By a similar argument to the sieve of sundaram, we can sieve out divisors of form $(p_1-1) k_m r_m+k_m+r_m$ because multiplying by $(p_1-1)$ and adding 1 gives us a factorization, which if $k_m,r_m>0$ is non-trivial ( namely $(k_m(p_1-1)+1)(r,_m(p_1-1)+1)$ )