SQLite loop statements?

Apparently the looping construct in SQLite is the WITH RECURSIVE clause.
That documentation link has sample count-to-ten code, a Mandelbrot set plotter, and a Sudoku puzzle solver, all in pure SQL.
Here’s an SQLite query that computes the Fibonacci sequence to give you a feel for it:

sqlite> WITH RECURSIVE
   ...>   fibo (curr, next)
   ...> AS
   ...>   ( SELECT 1,1
   ...>     UNION ALL
   ...>     SELECT next, curr+next FROM fibo
   ...>     LIMIT 100 )
   ...> SELECT group_concat(curr) FROM fibo;
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,10946,...

And here’s a Sieve of Eratosthenes:

begin transaction;

drop table if exists naturals;
create table naturals
( n integer unique primary key asc,
  isprime bool,
  factor integer);

with recursive
  nn (n)
as (
  select 2
  union all
  select n+1 as newn from nn
  where newn < 1e4
)
insert into naturals
select n, 1, null from nn;

insert or replace into naturals
  with recursive
    product (prime,composite)
  as (
    select n, n*n as sqr
      from naturals
      where sqr <= (select max(n) from naturals)
    union all
    select prime, composite+prime as prod
    from
      product
    where
      prod <= (select max(n) from naturals)
  )
select n, 0, prime
from product join naturals
  on (product.composite = naturals.n)
;
commit;