OFFSET
1,2
COMMENTS
For terms listed in the Data section: a(p^k) = a(p^k-1) + 1, where p prime (empirical observation). - Ilya Gutkovskiy, Dec 06 2020
From Chai Wah Wu, Dec 14 2020: (Start)
The above empirical observation is true.
Theorem: For prime p, a(p^k) = a(p^k-1)+1.
Proof: Since the singleton set {x} has harmonic mean x, a(n) >= a(n-1)+1.
Let S = {s_1,s_2,..,s_n} be a subset of {1,2,..,p^k} with n>1 elements such that s_n = p^k and let H be the harmonic mean of S. Let M = A003418(p^k) be the least common multiple of {1,2,..,p^k}. Then M = Wp^k where p does not divide W = A002944(p^k).
Let Q_i = M/s_i and Q = sum_i Q_i. This implies that Q_n = W and p divides Q_i for i < n.
H can be written as nM/Q. Since p does not divide W, this implies that p does not divide Q. Suppose H is an integer. Then this implies that Q divides nM/p^k = nW.
Note that s_i < s_n for i < n. This implies that Q_i > W for i < n, i.e. Q > nW, and this contradicts the fact that Q divides nW and thus H is not an integer.
Thus {p^k} is the only subset of {1,2,..,p^k} that includes p^k and have an integral Harmonic mean.
This concludes the proof.
(End)
LINKS
Eric W. Weisstein's World of Mathematics, Harmonic Mean
FORMULA
a(n) >= a(n-1)+1. For prime p, a(p^k) = a(p^k-1)+1. - Chai Wah Wu, Dec 14 2020
EXAMPLE
a(6) = 12 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {2, 6}, {3, 6}, {1, 3, 6}, {2, 3, 6}, {3, 4, 6} and {1, 2, 3, 6}.
PROG
(Python)
from itertools import chain, combinations
from fractions import Fraction
def powerset(s): # skip empty set
return chain.from_iterable(combinations(s, r) for r in range(1, len(s)+1))
def hm(s):
ss = sum(Fraction(1, i) for i in s)
return Fraction(len(s)*ss.denominator, ss.numerator)
def a(n):
return sum(hm(s).denominator==1 for s in powerset(range(1, n+1)))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Dec 06 2020
(Python)
from math import lcm
from itertools import combinations
def A339453(n):
m = lcm(*range(2, n+1))
return sum(1 for i in range(1, n+1) for d in combinations((m//i for i in range(1, n+1)), i) if m*i % sum(d) == 0) # Chai Wah Wu, Dec 02 2021
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Ilya Gutkovskiy, Dec 05 2020
EXTENSIONS
a(23)-a(29) from Michael S. Branicky, Dec 06 2020
a(30)-a(35) from Chai Wah Wu, Dec 08 2020
a(36)-a(39) from Chai Wah Wu, Dec 11 2020
a(40)-a(41) from Chai Wah Wu, Dec 19 2020
STATUS
approved