The sequence A003312 is the sequence with the "largest possible growth" while remaining decomposable into weight × level + jump.

[Rémi Eismann]

A003312: a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2): https://oeis.org/A003312

This sequence was originally defined in Popular Computing in 1974 by a sieve, as follows. Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every third term. Repeat, always crossing off every third term of those that remain. The numbers that are left form the sequence. The recurrence given here for the sequence was found by Colin Mallows. The problem asked for the 1000th term, and was unsolved for several years.
N. J. A. Sloane, Table of n, a(n) for n = 1..1000 (first 500 terms from T. D. Noe): https://oeis.org/A003312/b003312.txt

Best,

Rémi.