A339453 -id:A339453 - cp's OEIS frontend

A339454 Number of subsets of {1..n} whose root mean square is an integer.

1, 2, 3, 4, 5, 6, 9, 10, 15, 20, 29, 52, 87, 166, 311, 538, 943, 1682, 2915, 5054, 8905, 15904, 28533, 51826, 95191, 175402, 325777, 607542, 1134191, 2128922, 3986433, 7485522, 14065135, 26446388, 49796025, 93920770, 177470237, 335780796, 636883269, 1209603646

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(9) = 15 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9}, {1, 7}, {1, 5, 7}, {1, 3, 5, 8, 9}, {3, 4, 5, 7, 9}, {1, 3, 5, 6, 8, 9} and {3, 4, 5, 6, 7, 9}.

Max Alekseyev, Table of n, a(n) for n = 1..100
Eric W. Weisstein's World of Mathematics, Root-Mean-Square

Cf. A051293, A240089, A326027, A339453.

from functools import lru_cache
from sympy.ntheory.primetest import is_square
def cond(sos, c): return c > 0 and sos%c == 0 and is_square(sos//c)
@lru_cache(maxsize=None)
def b(n, sos, c):
    if n == 0: return int(cond(sos, c))
    return b(n-1, sos, c) + b(n-1, sos+n*n, c+1)
a = lambda n: b(n, 0, 0)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Oct 06 2022

a(n) = A357415(n) + A357416(n). - Max Alekseyev, Mar 25 2025

a(23)-a(40) from Alois P. Heinz, Dec 05 2020

A339665 Number of nonempty subsets of divisors of n whose harmonic mean is an integer.

Original entry on oeis.org

1, 2, 2, 3, 2, 9, 2, 4, 3, 4, 2, 17, 2, 4, 6, 5, 2, 19, 2, 10, 4, 4, 2, 37, 3, 4, 4, 12, 2, 45, 2, 6, 4, 4, 4, 57, 2, 4, 4, 28, 2, 29, 2, 6, 16, 4, 2, 85, 3, 6, 4, 6, 2, 35, 4, 23, 4, 4, 2, 301, 2, 4, 6, 7, 4, 28, 2, 6, 4, 19, 2, 255, 2, 4, 10, 6, 4, 20, 2, 61

Offset: 1

Ilya Gutkovskiy, Dec 11 2020

nonn

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

Chai Wah Wu, Table of n, a(n) for n = 1..3000
Eric Weisstein's World of Mathematics, Harmonic Mean
Index entries for sequences related to divisors of numbers

Cf. A027750, A100587, A339453, A339663, A339664, A339666.

a[n_] := Count[Subsets[Divisors[n]], ?(Length[#] > 0 && IntegerQ[HarmonicMean[#]] &)]; Array[a, 100] (* _Amiram Eldar, Nov 09 2021 *)

h(s, d) = #s/sum(k=1, #s, 1/d[s[k]]);
a(n) = my(d=divisors(n), nb=0); forsubset(#d, s, if (#s && (denominator(h(s, d))==1), nb++)); nb; \\ Michel Marcus, Dec 15 2020

from itertools import combinations
from sympy import divisors
def A339665(n):
    ds = tuple(divisors(n, generator=True))
    return sum(sum(1 for d in combinations(ds,i) if n*i % sum(d) == 0) for i in range(1,len(ds)+1)) # Chai Wah Wu, Nov 09 2021

A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

Original entry on oeis.org

1, 1, 2, 2, 3, 5, 6, 6, 7, 9, 10, 10, 11, 13, 26, 26, 27, 45, 46, 74, 93, 99, 100, 162, 163, 165, 166, 458, 459, 865, 866, 866, 1647, 1669, 2724

Offset: 1

Ilya Gutkovskiy, Sep 27 2022

			a(11) = 10 subsets: {1}, {3}, {5}, {7}, {9}, {11}, {2, 6}, {2, 3, 6}, {3, 6, 10} and {3, 5, 6, 10}.

Cf. A339453, A357355, A357412, A357413, A357415.

from fractions import Fraction
from functools import lru_cache
def cond(s, c): h = c/s; return h.denominator == 1 and h.numerator&1
@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) + 1 for prime p > 2. - Michael S. Branicky, Sep 30 2022

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

A362802 a(n) is the number of ways in which the set of divisors of n can be partitioned into disjoint parts, all of length > 1 and with integer harmonic mean.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 4, 0, 0, 0, 1, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 15, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 175, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 78, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 188, 0

Offset: 1

Amiram Eldar, May 04 2023

nonn

			 n  a(n)  partitions
==  ====  ==========
 6     1  {{1, 2, 3, 6}}
12     1  {{1, 2, 3, 6}, {4, 12}}
24     4  {{1, 2, 3, 6}, {4, 8, 12, 24}}, {{1, 2, 4, 8, 12, 24}, {3, 6}},
          {{1, 3, 6}, {2, 4, 8, 12, 24}}, {{1, 2, 3, 6}, {4, 12}, {8, 24}}

Cf. A339453, A339665, A362801, A362803 (indices of records).

harmQ[s_] := AllTrue[s, Length[#] > 1 && IntegerQ[HarmonicMean[#]] &]; a[n_] := Module[{d = Divisors[n], r}, r = ResourceFunction["SetPartitions"][d]; Count[r, _?harmQ]]; Array[a, 119]

a(A362801(n)) > 0.

A378844 Number of subsets of {1..n} whose arithmetic and harmonic means are both integers.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 13, 14, 18, 19, 21, 27, 28, 29, 48, 49, 71, 75, 78, 79, 103, 104, 105, 106, 203, 204, 325, 326, 327, 530, 533, 795, 1198, 1199, 1204

Offset: 1

Ilya Gutkovskiy, Dec 09 2024

			a(6) = 8 subsets: {1}, {2}, {3}, {4}, {5}, {6}, {2, 6} and {1, 2, 3, 6}.

Cf. A051293, A326643, A339453.

a[n_] := Count[Subsets[Range[n]], ?(# != {} && IntegerQ[Mean[#]] && IntegerQ[HarmonicMean[#]] &)]; Array[a, 20] (* _Amiram Eldar, Dec 10 2024 *)

a(n) = {my(c = 0); forsubset(n, s, if(#s && !(vecsum(Vec(s)) % #s) && denominator(#s/sum(i=1,#s,1/s[i])) == 1, c++)); c;} \\ Amiram Eldar, Dec 10 2024

from math import lcm
from itertools import combinations
def A378844(n):
    m = lcm(*range(2,n+1))
    return sum(1 for l in range(1,n+1) for c in combinations(range(1,n+1),l) if not (sum(c)%l or l*m%sum(m//d for d in c))) # Chai Wah Wu, Dec 12 2024

a(p^k) = a(p^k-1)+1 for p prime (see A339453). - Chai Wah Wu, Dec 12 2024

a(25)-a(29) from Amiram Eldar, Dec 10 2024
a(30)-a(33) from Chai Wah Wu, Dec 12 2024
a(34)-a(38) from Chai Wah Wu, Dec 13 2024

cp's OEIS Frontend

A339454 Number of subsets of {1..n} whose root mean square is an integer.

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A339665 Number of nonempty subsets of divisors of n whose harmonic mean is an integer.

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A357411 Number of nonempty subsets of {1..n} whose elements have an odd harmonic mean.

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

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A362802 a(n) is the number of ways in which the set of divisors of n can be partitioned into disjoint parts, all of length > 1 and with integer harmonic mean.

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A378844 Number of subsets of {1..n} whose arithmetic and harmonic means are both integers.

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Mathematica

PARI

Python

Formula

Extensions