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

2,1

Comments

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

Links

Table of n, a(n) for n=2..80.

Example

   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.

Prog

(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)

Crossrefs

Cf. A080577, A000586.

Sequence in context: A254862 A340641 A322235 * A346477 A337583 A172984

Adjacent sequences: A385704 A385705 A385706 * A385708 A385709 A385710

Keyword

nonn,tabf

Author

Jens Ahlström, Jul 07 2025

Status

approved