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

%I A382381 #20 Apr 07 2025 17:46:47
%S A382381 1,2,4,8,16,25,36,62,136,320,411,1208,1295,4179,5143,6380,31370,34425,
%T A382381 36094,213044,218759,306722
%N A382381 Lexicographically earliest sequence of distinct positive integers such that any two subsets with at least two terms have distinct variances.
%C A382381 Numbers k such that A381856(k) = 1.
%C A382381 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.
%C A382381 a(20) > 100000.
%o A382381 (Python)
%o A382381 from fractions import Fraction
%o A382381 from itertools import chain, combinations, count, islice
%o A382381 def powerset(s): # skipping empty set
%o A382381     return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
%o A382381 def agen(): # generator of terms
%o A382381     an, alst, vset = 1, [1], set()
%o A382381     while True:
%o A382381         yield an
%o A382381         P = list(powerset(alst))
%o A382381         Xlst, X2lst = [sum(s) for s in P], [sum(si**2 for si in s) for s in P]
%o A382381         for k in count(an+1):
%o A382381             ok, vnew = True, set()
%o A382381             for i, s in enumerate(P):
%o A382381                 mu, X2 = Fraction(Xlst[i] + k, len(s)+1), X2lst[i] + k**2
%o A382381                 v = Fraction(X2, len(s)+1) - mu**2
%o A382381                 if v in vset or v in vnew:
%o A382381                     ok = False
%o A382381                     break
%o A382381                 else:
%o A382381                     vnew.add(v)
%o A382381             if ok:
%o A382381                 break
%o A382381         an = k
%o A382381         vset |= vnew
%o A382381         alst.append(an)
%o A382381 print(list(islice(agen(), 13))) # _Michael S. Branicky_, Mar 31 2025
%Y A382381 Cf. A138857, A260873, A381856, A382382, A382383.
%K A382381 nonn,hard,more
%O A382381 1,2
%A A382381 _Pontus von Brömssen_, Mar 23 2025
%E A382381 a(20)-a(21) from _Michael S. Branicky_, Mar 31 2025
%E A382381 a(22) from _Michael S. Branicky_, Apr 07 2025