A382382 -id:A382382 - cp's OEIS frontend

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

Pontus von Brömssen, Mar 23 2025

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.

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

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

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

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

A382383 Number of distinct variances of nonempty subsets of {1, ..., n}.

Original entry on oeis.org

0, 1, 2, 4, 7, 13, 23, 40, 68, 124, 208, 368, 559, 918, 1352, 2017, 2891, 4122, 5506, 7458, 9623, 12620, 16125, 20626, 25401, 31513, 38587, 47244, 56592, 68021, 80503, 95859, 112137, 131986, 153353, 178434, 205627, 236266, 269884, 307167, 346844, 394924, 445797, 501739

Offset: 0

Pontus von Brömssen, Mar 23 2025

nonn

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.

			For n = 4, the following a(4) = 7 variances occur for subsets of {1, 2, 3, 4}:
   variance | corresponding subsets
   ---------+----------------------
       0    | {1}, {2}, {3}, {4}
      1/4   | {1,2}, {2,3}, {3,4}
      2/3   | {1,2,3}, {2,3,4}
       1    | {1,3}, {2,4}
      5/4   | {1,2,3,4}
     14/9   | {1,2,4}, {1,3,4}
      9/4   | {1,4}

Cf. A005418, A135342, A208531, A382381, A382382.

from fractions import Fraction
def A382383_lst(n):
    s,lst=set(),[0]
    for k in range(n):
        s|={ (x+k,x2+k**2,l+1 ) for (x,x2,l) in s }
        s.add( (k,k**2,1) )
        lst.append(len({ Fraction(x2,l) - Fraction(x,l)**2 for (x,x2,l) in s }))
    return lst # Bert Dobbelaere, Apr 06 2025

a(n) <= 1 + Sum_{k=1..n-1} A005418(k). The smallest positive n for which strict inequality holds is n = 7. This is because there exist subsets of {1, ..., 7} which are not translates or reflections of each other, but nonetheless have the same variance. For example, {1,5}, {1,3,4,5,7}, and {1,2,3,4,5,6,7} all have variance 4, and {1,2,6} and {1,2,3,5,6,7} both have variance 14/3.

a(34)-a(43) from Bert Dobbelaere, Apr 06 2025

A382832 Least k such that there exist two distinct subsets of {0, ..., k-1} with the same sum of m-th powers for 0 <= m <= n.

Original entry on oeis.org

2, 4, 7, 12, 16, 23, 31

Offset: 0

Pontus von Brömssen, Apr 10 2025

Two such sets must have the same size, since the exponent m = 0 is allowed (with the usual convention that 0^0 = 1).

a(n) is the smallest k such that A382833(k,n) < 2^k.

			  n | a(n) | subsets with the same sums of powers
  --+------+-------------------------------------
  0 |   2  | {0}, {1}
  1 |   4  | {0,3}, {1,2}
  2 |   7  | {0,4,5}, {1,2,6}
  3 |  12  | {0,4,7,11}, {1,2,9,10}
  4 |  16  | {0,5,6,7,13,14}, {1,2,8,9,10,15}
  5 |  23  | {0,5,6,16,17,22}, {1,2,10,12,20,21}
  6 |  31  | {0,5,6,9,16,17,18,22,28,29}, {1,2,8,12,13,14,21,24,25,30}
For n = 3, the two subsets {0,4,7,11} and {1,2,9,10} of {0, ..., 11} have the same sum of m-th powers for 0 <= m <= 3: 0^0+4^0+7^0+11^0 = 1^0+2^0+9^0+10^0 = 4, 0^1+4^1+7^1+11^1 = 1^1+2^1+9^1+10^1 = 22, 0^2+4^2+7^2+11^2 = 1^2+2^2+9^2+10^2 = 186, 0^3+4^3+7^3+11^3 = 1^3+2^3+9^3+10^3 = 1738. There are no such subsets of {0, ..., 10}, so a(3) = 12.

Cf. A382382, A382833.

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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.

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A382383 Number of distinct variances of nonempty subsets of {1, ..., n}.

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A382832 Least k such that there exist two distinct subsets of {0, ..., k-1} with the same sum of m-th powers for 0 <= m <= n.

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