cp's OEIS Frontend

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.

%I A385707 #26 Jul 23 2025 16:15:25
%S A385707 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,
%T A385707 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,
%U A385707 11,5,3,17,3,13,7,13,5,2,11,7,2,19,2,13,5,3
%N 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.
%C A385707 A table of partitions of n into distinct prime parts in graded reverse lexicographic ordering.
%e A385707    r    n \ k  1    2    3    4    5
%e A385707   -----------------------------------
%e A385707    1    2    [ 2];
%e A385707    2    3    [ 3];
%e A385707    3    5    [ 5], [3,   2];
%e A385707    4    7    [ 7], [5,   2];
%e A385707    5    8    [ 5,   3];
%e A385707    6    9    [ 7,   2];
%e A385707    7   10    [ 7,   3], [5,   3,   2];
%e A385707    8   11    [11];
%e A385707    9   12    [ 7,   5], [7,   3,   2];
%e A385707    ...
%e A385707 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.
%o A385707 (Python)
%o A385707 from sympy.utilities.iterables import partitions
%o A385707 from sympy import isprime
%o A385707 res = []
%o A385707 for n in range(22):
%o A385707     for p in partitions(n):
%o A385707         for i, f in p.items():
%o A385707             if not isprime(i) or f>1:
%o A385707                 break
%o A385707         else:
%o A385707             res.extend(list(p.keys()))
%o A385707 print(res)
%Y A385707 Cf. A080577, A000586.
%K A385707 nonn,tabf
%O A385707 2,1
%A A385707 _Jens Ahlström_, Jul 07 2025