A382381
Lexicographically earliest sequence of distinct positive integers such that any two subsets with at least two terms have distinct variances.
Original entry on oeis.org
1, 2, 4, 8, 16, 25, 36, 62, 136, 320, 411, 1208, 1295, 4179, 5143, 6380, 31370, 34425, 36094, 213044, 218759, 306722
Offset: 1
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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
A138858
a(n) = least square such that the subsets of {a(1),...,a(n)} sum to 2^n different values.
Original entry on oeis.org
1, 4, 9, 16, 36, 81, 144, 324, 625, 1156, 2401, 4900, 9801, 19600, 39204, 78400, 156816, 313600, 627264, 1254400, 2509056, 5022081, 10042561, 20088324, 40182921, 80371225, 160731684, 321484900, 642977449, 1285939600, 2571909796, 5143901841, 10287842041, 20575607364, 41151368164, 82303003225, 164606284089, 329212012900, 658425136356, 1316850346681, 2633700545424, 5267401386724
Offset: 1
Up to a(4)=16, we have a(n)=n^2.
But since 5^2=25=9+16 is already represented as sum of earlier terms, this is excluded, while a(5)=6^2=36 has the required property.
Obviously, any square larger to the sum of all preceding terms leads to enough new terms, thus a(n) <= floor( sqrt( sum(a(k),k=1..n-1))+1)^2.
But in contrast to A064934, such a simple formula (with equality) cannot be used here:
a(7)=12^2=144 < 147=sum(a(k),k<7) and also a(10)=sum(a(k),k<10)-84.
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{s=1;p=0; for( n=1,20, until( !bitand( s, s>>(p^2) ), p++); s+=s<<(p^2); print1( p^2,","))}
A138856
Numbers such that all subsets of {prime(a(1)), ..., prime(a(n))} have a different sum.
Original entry on oeis.org
1, 2, 4, 5, 10, 16, 28, 47, 83, 147, 267, 481, 882, 1621, 2997, 5578, 10428, 19560, 36849, 69649, 131983, 250841, 477992, 912662, 1746404, 3347928, 6429526, 12366247, 23820901, 45947255, 88742186, 171594310, 332169919, 643674781, 1248523100, 2423948034
Offset: 1
-
{s=1;p=0; for( n=1,20, until( !bitand( s, s>>prime(p++) ),); s+=s<
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