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

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

1,2

Links

Table of n, a(n) for n=1..38.

Formula

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

Example

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

Mathematica

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

Prog

(PARI) 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
(Python)
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

Crossrefs

Cf. A051293, A326643, A339453.

Sequence in context: A047369 A004827 A155941 * A046892 A068406 A276878

Adjacent sequences: A378841 A378842 A378843 * A378845 A378846 A378847

Keyword

nonn,more

Author

Ilya Gutkovskiy, Dec 09 2024

Extensions

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

Status

approved