A091911 -id:A091911 - cp's OEIS frontend

A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 19, 21, 24, 25, 26, 27, 29, 31, 35, 37, 39, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 53, 54, 60, 61, 70, 73, 75, 77, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 94, 96, 97, 99, 101, 102, 105, 106, 107, 108, 110, 114, 115

Offset: 1

R. K. Guy, Feb 08 2004

nonn

The equation n = m*tau(m)/sigma(m) has an integer solution m.
Here tau(n) (A000005) is the number of divisors of n and sigma(n) is the sum of the divisors of n (A000203).
A001600 sorted in order.
The Mersenne exponents (A000043) are in this sequence because the even perfect numbers, 2^(p-1)*(2^p-1) where p is in A000043, are all harmonic numbers (A001599) with harmonic mean of divisors p. - Amiram Eldar, Apr 15 2024

For further references see A001599.

T. Goto and S. Shibata, All numbers whose positive divisors have integral harmonic mean up to 300, Math. Comput. 73 (2004), 475-491.

Values of m are in A091911.
Complement of A157849.
Cf. A000043, A001599, A001600, A090758, A090759, A090760, A090761, A090762.

f[n_] := (n*DivisorSigma[0, n]/DivisorSigma[1, n]); a = Table[ 0, {120}]; Do[ b = f[n]; If[ IntegerQ[b] && b < 121 && a[[b]] == 0, a[[b]] = n], {n, 1, 560000000}]; Select[ Range[120], a[[ # ]] > 0 &] (* Robert G. Wilson v, Feb 14 2004 *)

More terms from Robert G. Wilson v, Feb 14 2004

A327054 a(n) is the smallest number m such that the antiharmonic mean of the divisors is n, or 0 if no such m exists.

Original entry on oeis.org

1, 0, 4, 0, 0, 0, 9, 0, 0, 0, 16, 0, 20, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 0, 0, 0, 0, 0, 0, 0, 49, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 81, 0, 100, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 144, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Jaroslav Krizek, Oct 06 2019

nonn

a(n) = the smallest number m such that sigma_2(n) / sigma_1(n) = A001157(m) / A000203(m) = n, or 0 if no such m exists.
Zeros occur if n is not in A176799.
See A000290, A091911 and A162538 for like sequences for geometric, arithmetic and harmonic means of the divisors.

			a(3) = 4 because 4 is the smallest number m with sigma_2(m) / sigma_1(m) = 3; sigma_2(4) / sigma_1(4) = 21 / 7 = 3.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A020487, A176797, A176799, A176800.

Cf. A000290, A004394, A091911, A162538.

A327054:=func; [A327054(n): n in[1..100]];

# This uses the b-file for A004394
# See comment at A176799
K:= 100: # to get terms <= K
M:= 36 * K^2/Pi^4:
for i from 1 while A004394[i] < M do od:
r:= numtheory:-sigma(A004394[i])/A004394[i]:
V:= Vector(K):
for m from 1 to r*K do
  F:= numtheory:-divisors(m);
  v:= add(d^2, d=F)/add(d, d=F);
  if v::integer and v <= K and V[v] = 0 then V[v]:= m fi;
od:
convert(V,list); # Robert Israel, Sep 05 2024

cp's OEIS Frontend

A090240 Numbers which occur as the harmonic mean of the divisors of m for some m.

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