cp's OEIS Frontend

A385675 For positive integers n, a multiset A of positive integers is called "n-good" if for any number 1 <= i <= n, we can find a submultiset B of A such that the sum of B is equal to i. a(n) is the number of minimal n-good multisets.

%I A385675 #20 Aug 18 2025 16:23:46
%S A385675 1,2,3,7,9,18,27,47,62,101,140,226,301,437,579,838,1077,1525,1985,
%T A385675 2721,3470,4674,5899,7843,9773,12703,15803,20431,25129,32167,39519,
%U A385675 49982,60928,76373,92537,115313,138969,171372,205847,252604,301444,367890,438145,531202,630209
%N A385675 For positive integers n, a multiset A of positive integers is called "n-good" if for any number 1 <= i <= n, we can find a submultiset B of A such that the sum of B is equal to i. a(n) is the number of minimal n-good multisets.
%C A385675 An n-good multiset A is minimal if it is impossible to get another n-good multiset by deleting one element from A.
%C A385675 An n-good multiset must contain 1 and have a sum of elements >= n. - _Michael S. Branicky_, Aug 13 2025
%e A385675 For n = 6, {1, 2, 3} and {1, 1, 3, 6} are minimal n-good multisets.
%o A385675 (Python) # for illustrative purposes
%o A385675 from functools import cache
%o A385675 from itertools import chain, combinations, combinations_with_replacement as cwr
%o A385675 def f(t): return tuple(e for e in t if e != 0)
%o A385675 def powerset(s): return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))
%o A385675 @cache
%o A385675 def good(n, A): return 1 in A and sum(A) >= n and set(range(1, n+1))-set(sum(B) for B in powerset(A)) == set()
%o A385675 def a(n): return sum(good(n, A) and all(not good(n, A[:i]+A[i+1:]) for i in range(len(A))) for A in map(f, cwr(range(n+1), n)))
%o A385675 print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, Aug 13 2025
%Y A385675 Cf. A126796.
%K A385675 nonn
%O A385675 1,2
%A A385675 _Yifan Xie_, Aug 05 2025