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.

1, 2, 3, 7, 9, 18, 27, 47, 62, 101, 140, 226, 301, 437, 579, 838, 1077, 1525, 1985, 2721, 3470, 4674, 5899, 7843, 9773, 12703, 15803, 20431, 25129, 32167, 39519, 49982, 60928, 76373, 92537, 115313, 138969, 171372, 205847, 252604, 301444, 367890, 438145, 531202, 630209

Offset: 1

Author

Yifan Xie, Aug 05 2025

Keywords

• nonn

Comments

An n-good multiset A is minimal if it is impossible to get another n-good multiset by deleting one element from A.

An n-good multiset must contain 1 and have a sum of elements >= n. - Michael S. Branicky, Aug 13 2025

Examples

			For n = 6, {1, 2, 3} and {1, 1, 3, 6} are minimal n-good multisets.
		

Crossrefs

Cf. A126796.

Programs

Python

# for illustrative purposes
from functools import cache
from itertools import chain, combinations, combinations_with_replacement as cwr
def f(t): return tuple(e for e in t if e != 0)
def powerset(s): return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))
@cache
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()
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)))
print([a(n) for n in range(1, 11)]) # Michael S. Branicky, Aug 13 2025