The smallest parts function 
counts, with multiplicity,
the occurrences of the smallest part in all partitions
of a positive integer 
. For example, the partitions
of 4 contribute 
. For 
, 2, ..., the values are 1, 3, 5, 10, 14, 26, 35,
57, 80, 119, ... (OEIS A092269). Set 
.
For a prime number 
, let 
be the integer with 
and 
. A canonical congruence
prime is one for which
 |
for every nonnegative integer 
. Rolen (2026) proved that these primes
are precisely 5, 7, and 13. The corresponding progressions are 
, 
,
and 
. This classifies the canonical prime progressions,
not all possible congruences for the function.
Rolen (2026) reports AI assistance in developing the proof and an AxiomProver formalization in Lean.
." J. Reine Angew. Math. 624, 133-142, 2008.