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A339453 Number of subsets of {1..n} whose harmonic mean is an integer. 2
1, 2, 3, 4, 5, 12, 13, 14, 15, 18, 19, 26, 27, 30, 53, 54, 55, 100, 101, 180, 203, 210, 211, 378, 379, 382, 383, 1092, 1093, 2020, 2021, 2022, 3933, 3956, 6473, 10226, 10227, 10266, 10561, 20948, 20949 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..41.

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

CROSSREFS

Cf. A002944, A003418, A051293, A067540, A246655, A326027, A339454.

Sequence in context: A066574 A240304 A325684 * A281597 A039007 A050745

Adjacent sequences:  A339450 A339451 A339452 * A339454 A339455 A339456

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

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Last modified October 20 15:49 EDT 2021. Contains 348111 sequences. (Running on oeis4.)