cp's OEIS Frontend

A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

%I A364346 #13 Oct 18 2023 04:44:21
%S A364346 1,1,1,1,2,3,2,4,4,5,5,8,9,11,11,16,16,20,20,25,30,34,38,42,50,58,64,
%T A364346 73,80,90,105,114,128,148,158,180,201,220,241,277,306,333,366,404,447,
%U A364346 497,544,592,662,708,797,861,954,1020,1131,1226,1352,1456,1600
%N A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.
%e A364346 The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):
%e A364346   1   2   3   4    5    6    7    8    9     A    B     C     D     E
%e A364346               31   32   51   43   53   54    64   65    75    76    86
%e A364346                    41        52   62   72    73   74    93    85    95
%e A364346                              61   71   81    82   83    A2    94    A4
%e A364346                                        531   91   92    B1    A3    B3
%e A364346                                                   A1    543   B2    C2
%e A364346                                                   641   732   C1    D1
%e A364346                                                   731   741   652   851
%e A364346                                                         831   751   932
%e A364346                                                               832   941
%e A364346                                                               931   A31
%t A364346 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,15}]
%o A364346 (Python)
%o A364346 from collections import Counter
%o A364346 from itertools import combinations_with_replacement
%o A364346 from sympy.utilities.iterables import partitions
%o A364346 def A364346(n): return sum(1 for p in partitions(n) if max(p.values(),default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # _Chai Wah Wu_, Sep 20 2023
%Y A364346 For subsets of {1..n} we have A007865 (sum-free sets), differences A288728.
%Y A364346 For sums of any length > 1 we have A364349, non-strict A237667.
%Y A364346 The complement is counted by A363226, non-strict A363225.
%Y A364346 The non-strict version is A364345, ranks A364347, complement A364348.
%Y A364346 A000041 counts partitions, strict A000009.
%Y A364346 A008284 counts partitions by length, strict A008289.
%Y A364346 A236912 counts sum-free partitions not re-using parts, complement A237113.
%Y A364346 A323092 counts double-free partitions, ranks A320340.
%Y A364346 Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.
%K A364346 nonn
%O A364346 0,5
%A A364346 _Gus Wiseman_, Jul 22 2023