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.
