A397311 a(1) = 0; for n > 1, a(n) is the least integer > a(n-1) such that distinct multisets of 10 terms from a(1),...,a(n) have distinct sums.

0, 1, 11, 111, 616, 4448, 23614, 100943, 430134, 1529622, 5273780, 18181274

Offset

1,3

Comments

This is a greedy B_10 sequence beginning with 0.

Links

Table of n, a(n) for n=1..12.

Adam Reichert, Python program using sieving; faster but more memory-intensive

Example

a(3)!=10 because 10+0+0+0+0+0+0+0+0+0 = 1+1+1+1+1+1+1+1+1+1.

Prog

(Python)
import itertools
def greedy_bh(h, start):
    terms, sums = [], set()
    def test_candidate(candidate):
        new_sums = []
        for copies in range(1, h + 1):
            for combo in itertools.combinations_with_replacement(terms, h - copies):
                candidate_sum = sum(combo) + candidate * copies
                # No need to check candidate_sum against the other values in
                # `new_sums`: any collision between two new sums, one with r
                # copies and one with s (r < s), reduces to a new-vs-old
                # collision at s-r copies, which is caught earlier.
                if candidate_sum in sums:
                    return None
                new_sums.append(candidate_sum)
        return new_sums
    for candidate in itertools.count(start):
        if (candidate_sums := test_candidate(candidate)) is not None:
            yield candidate
            terms.append(candidate)
            sums.update(candidate_sums)
a397311_list = list(itertools.islice(greedy_bh(10, 0), 5))

Crossrefs

Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365304, A365305.

Sequence in context: A062095 A055657 A049121 * A143573 A248039 A244204

Adjacent sequences: A397297 A397301 A397304 * A397312 A397313 A397314

Keyword

nonn,more,new

Author

Adam Reichert, Jun 20 2026

Status

approved