Running least common multiples: distinct values and multiplicities

The sums over distinct running LCMs

We record the ordinary arguments for general finite prime supports. Their assertions, including the block and finite-state criteria below, are presented as remarks because this part has not been formalised. The complete proofs are retained. The five-map theorem in Section 2 has a separate formal proof. No argument here gives transcendence for more than two primes.

Let $P$ be a finite set of at least two primes, with least element $p$, and put ${\mathrm{H}}_P(t)=\prod _{r\in P}{r}^{\lfloor {\log }_{r}t\rfloor }$. The proofs of Theorem 3.1 and Propositions 3.2 and 3.3 apply unchanged to any finite set of primes: the running least common multiple of the $P$-smooth integers up to $t$ is ${\mathrm{H}}_P(t)$, it changes only at the prime powers $r^n$ ($r\in P$, $n\ge 1$), where it is multiplied by $r$, and distinct prime powers are distinct numbers. List the prime powers as $t_1<t_2<\cdots$ and let $q_k$ be the prime of $t_k$. The distinct running values are then $1$ and ${\mathrm{H}}_P(t_k)=q_1\cdots q_k$, so

This is the sum ${\mathrm{\Sigma }}^{\prime }$ of Erdős’s letter [3], which counts each distinct value of $[a_1,\dots ,a_n]$ once. For $P=\{2,3,5\}$ the prime powers $2,3,4,5,8,9,16,25,27,32$ give the following terms, including the initial $1$

Numerically, ${\mathcal{D}}_{\{2,3,5\}}=1.77961698329474905045919064931698327\dots$. Put $Q_k=q_1\cdots q_k={\mathrm{H}}_P(t_k)$.

Proof for an arbitrary finite prime set

Blocks and affine maps

A decoding criterion for every finite set of primes

Certificates for three and four primes

Repeated tails and prime-power subseries