A382381 Lexicographically earliest sequence of distinct positive integers such that any two subsets with at least two terms have distinct variances.

1, 2, 4, 8, 16, 25, 36, 62, 136, 320, 411, 1208, 1295, 4179, 5143, 6380, 31370, 34425, 36094, 213044, 218759, 306722

Offset: 1

Author

Pontus von Brömssen, Mar 23 2025

Keywords

• nonn
• hard
• more

Comments

Numbers k such that A381856(k) = 1.

The variance of a nonempty set X is (Sum_{x in X} (x-m)^2)/|X|, where m is the average of X and |X| is the size of X.

a(20) > 100000.

Crossrefs

Cf. A138857, A260873, A381856, A382382, A382383.

Programs

Python

from fractions import Fraction
from itertools import chain, combinations, count, islice
def powerset(s): # skipping empty set
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
def agen(): # generator of terms
    an, alst, vset = 1, [1], set()
    while True:
        yield an
        P = list(powerset(alst))
        Xlst, X2lst = [sum(s) for s in P], [sum(si**2 for si in s) for s in P]
        for k in count(an+1):
            ok, vnew = True, set()
            for i, s in enumerate(P):
                mu, X2 = Fraction(Xlst[i] + k, len(s)+1), X2lst[i] + k**2
                v = Fraction(X2, len(s)+1) - mu**2
                if v in vset or v in vnew:
                    ok = False
                    break
                else:
                    vnew.add(v)
            if ok:
                break
        an = k
        vset |= vnew
        alst.append(an)
print(list(islice(agen(), 13))) # Michael S. Branicky, Mar 31 2025

Extensions

a(20)-a(21) from Michael S. Branicky, Mar 31 2025

a(22) from Michael S. Branicky, Apr 07 2025