A357411 -id:A357411 - cp's OEIS frontend

A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

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

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

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)

			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}.

Eric W. Weisstein's World of Mathematics, Harmonic Mean

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

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

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

a(n) >= a(n-1)+1. For prime p, a(p^k) = a(p^k-1)+1. - Chai Wah Wu, Dec 14 2020

a(n) = A357411(n) + A357412(n). - Max Alekseyev, Feb 26 2025

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

A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 3, 4, 4, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 19, 19, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 39, 39, 40, 40, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 90, 90, 91, 91, 92, 92

Offset: 0

Ilya Gutkovskiy, Sep 27 2022

nonn

The geometric mean of a subset such as in name must be an odd number in {1..n} which might ease the search for terms. - David A. Corneth, Sep 29 2022

			a(9) = 7 subsets: {1}, {3}, {5}, {7}, {9}, {1, 9} and {1, 3, 9}.

Max Alekseyev, Table of n, a(n) for n = 0..256

Cf. A001055, A326027, A357355, A357411, A357414, A357415.

from functools import lru_cache
from sympy import integer_nthroot
def cond(p, c): r, b = integer_nthroot(p, c); return b and r&1
@lru_cache(maxsize=None)
def b(n, p, c):
    if n == 0: return int (c > 0 and cond(p, c))
    return b(n-1, p, c) + b(n-1, p*n, c+1) if n&1 else b(n-1, p, c)
@lru_cache(maxsize=None)
def a(n): return b(n, 1, 0) if n&1 else b(n-1, 1, 0) if n else 0
print([a(n) for n in range(41)]) # Michael S. Branicky, Sep 29 2022

a(2*n-1) = a(2*n) for n >= 1. - David A. Corneth, Sep 29 2022

a(n) = A326027(n) - A357414(n). - Max Alekseyev, Mar 01 2025

a(24)-a(34) from Michael S. Branicky, Sep 29 2022
a(35)-a(70) from David A. Corneth, Sep 29 2022
a(0) prepended and terms a(71) onward added by Max Alekseyev, Mar 06 2025

A357415 Number of nonempty subsets of {1..n} whose elements have an odd root mean square.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 6, 7, 9, 16, 26, 41, 85, 142, 254, 461, 825, 1454, 2506, 4535, 7987, 14352, 26178, 47861, 87945, 162486, 304864, 565217, 1064529, 1992628, 3742934, 7034489, 13214869, 24924676, 46926388, 88812537, 167903969, 318619708, 604909434, 1150800393

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

nonn

			a(10) = 9 subsets: {1}, {3}, {5}, {7}, {9}, {1, 7}, {1, 5, 7}, {2, 3, 6, 8, 9, 10} and {2, 3, 6, 7, 8, 9, 10}.

Max Alekseyev, Table of n, a(n) for n = 1..100

Cf. A339454, A357355, A357411, A357413, A357416.

a(n) = A339454(n) - A357416(n).

a(24)-a(41) from Alois P. Heinz, Sep 27 2022

A357412 Number of nonempty subsets of {1..n} whose elements have an even harmonic mean.

Original entry on oeis.org

0, 1, 1, 2, 2, 7, 7, 8, 8, 9, 9, 16, 16, 17, 27, 28, 28, 55, 55, 106, 110, 111, 111, 216, 216, 217, 217, 634, 634, 1155, 1155, 1156, 2286, 2287, 3749

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

			a(11) = 9 subsets: {2}, {4}, {6}, {8}, {10}, {3, 6}, {1, 3, 6}, {3, 4, 6} and {1, 2, 3, 6}.

Cf. A339453, A357356, A357411, A357414, A357416.

from fractions import Fraction
from functools import lru_cache
def cond(s, c): h = c/s; return h.denominator == 1 and h.numerator&1 == 0
@lru_cache(maxsize=None)
def b(n, s, c):
    if n == 0: return int (c > 0 and cond(s, c))
    return b(n-1, s, c) + b(n-1, s+Fraction(1, n), c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Sep 29 2022

a(p) = a(p-1) for prime p > 2. - Michael S. Branicky, Sep 30 2022

a(24)-a(35) from Michael S. Branicky, Sep 30 2022

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A339453 Number of subsets of {1..n} whose harmonic mean is an integer.

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A357413 Number of nonempty subsets of {1..n} whose elements have an odd geometric mean.

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A357415 Number of nonempty subsets of {1..n} whose elements have an odd root mean square.

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Extensions

A357412 Number of nonempty subsets of {1..n} whose elements have an even harmonic mean.

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Author

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Examples

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Programs

Python

Formula

Extensions