wondering about multiplicative (not arithmetic) sequences of primes

Question

(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.

Answer 1

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.

Answer 2

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 $(k_m(p_1-1)+1)(r,_m(p_1-1)+1)$ )