A260873 -id:A260873 - 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

A380968 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same mean.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 3, 2, 4, 4, 5, 3, 1, 4, 5, 5, 6, 6, 7, 4, 6, 7, 2, 5, 8, 6, 3, 7, 1, 7, 5, 8, 8, 4, 9, 8, 9, 9, 10, 10, 6, 10, 9, 11, 11, 10, 11, 2, 8, 12, 11, 3, 7, 10, 12, 5, 12, 9, 11, 4, 13, 13, 14, 13, 12, 6, 14, 13, 14, 10, 15, 15, 16, 15, 11, 13

Offset: 1

Neal Gersh Tolunsky, Feb 09 2025

nonn

A260873 gives the indices of 1s in the sequence.
The longest run in the sequence has length 2.
No three equal terms will appear at indices in arithmetic progression.
For any value k, the distances between pairs of k will be distinct.

			a(7) = 3: a(7) cannot be 1 because i = 4; i = 1,7; and i = 1,4,7 would all have the same mean index 4. a(7) cannot be 2 because i = 6; i = 5,6,7; and i = 5,7 would have the same mean index 6. So a(7) = 3.
a(19) cannot be 1, 2, or 3. a(19) = 4 does not work either because i = 13,19 would have the same mean (namely 16) as i = 12,17,19. So a(19) = 5.

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Cf. A380783, A380751.

A381063 Lexicographically earliest sequence of positive integers such that each nonempty subset has a distinct geometric mean.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 17, 18, 19, 23, 29, 31, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 176, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251

Offset: 1

Neal Gersh Tolunsky, Feb 12 2025

nonn

Every prime occurs in the sequence.

Cf. A066720, A260873.

is(v, w, k) = my(f=factor(k), x, y, z); for(i=1, #f~, if(setsearch(v, f[i, 1]), listput(w, f[i, 1]))); w=Set(w); forsubset(#w, r, x=#r; y=k*prod(i=1, x, w[r[i]]); forsubset(#w, s, z=#s; if(y^z==prod(i=1, z, w[s[i]])^(x+1)&&z>0, return(0)))); 1;
lista(nn) = my(f, k=1, v=[1], w=List([1])); for(n=2, nn, while(!isprime(k++), if(is(v, w, k), f=factor(k); for(i=1, #f~, if(setsearch(v, f[i, 1]), listput(w, f[i, 1]))); listput(w, k); break)); v=concat(v, k)); v; \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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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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A380968 Lexicographically earliest sequence of positive integers such that for any value k, no two sets of one or more indices at which k occurs have the same mean.

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A381063 Lexicographically earliest sequence of positive integers such that each nonempty subset has a distinct geometric mean.

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