A385707 Irregular triangle T(r,k) read by rows in which row r lists the partitions into distinct primes of the r-th number having such partitions, r >= 1, k >= 1.

2, 3, 5, 3, 2, 7, 5, 2, 5, 3, 7, 2, 7, 3, 5, 3, 2, 11, 7, 5, 7, 3, 2, 13, 11, 2, 11, 3, 7, 5, 2, 13, 2, 7, 5, 3, 13, 3, 11, 5, 11, 3, 2, 17, 7, 5, 3, 2, 13, 5, 13, 3, 2, 11, 7, 11, 5, 2, 19, 17, 2, 11, 5, 3, 17, 3, 13, 7, 13, 5, 2, 11, 7, 2, 19, 2, 13, 5, 3

Offset: 2

Author

Jens Ahlström, Jul 07 2025

Keywords

• nonn
• tabf

Comments

A table of partitions of n into distinct prime parts in graded reverse lexicographic ordering.

Examples

			   r    n \ k  1    2    3    4    5
  -----------------------------------
   1    2    [ 2];
   2    3    [ 3];
   3    5    [ 5], [3,   2];
   4    7    [ 7], [5,   2];
   5    8    [ 5,   3];
   6    9    [ 7,   2];
   7   10    [ 7,   3], [5,   3,   2];
   8   11    [11];
   9   12    [ 7,   5], [7,   3,   2];
   ...
For n = 10 we can see that 10 is the 7th number having partitions into distinct primes so the 7th row of the triangle lists the two partitions that are the two ways to write 10 as a sum of distinct primes: 7 + 3 and 5 + 3 + 2.
		

Crossrefs

Cf. A080577, A000586.

Programs

Python

from sympy.utilities.iterables import partitions
from sympy import isprime
res = []
for n in range(22):
    for p in partitions(n):
        for i, f in p.items():
            if not isprime(i) or f>1:
                break
        else:
            res.extend(list(p.keys()))
print(res)