A099378 -id:A099378 - cp's OEIS frontend

A348695 a(n) is the least number k such that the denominator of the harmonic mean of the divisors of k is equal to n, or -1 if no such k exists.

1, 3, 2, 7, 24, 11, 4, 21, 10, 19, 258, 23, 9, 39, 8, 31, 402, 55, 37, 57, 26, 43, 3836, 47, 216, 99, 34, 124, 4844, 59, 16, 93, 86, 67, 76, 71, 73, 111, 125, 79, 978, 83, 7196, 129, 58, 411, 7868, 155, 52, 447, 101, 63, 1266, 107, 109, 372, 74, 519, 9884, 203

Offset: 1

Amiram Eldar, Oct 30 2021

nonn

			a(2) = 3 since the harmonic mean of the divisors of 3 is 3/2.
a(3) = 2 since the harmonic mean of the divisors of 2 is 4/3.

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A099377, A099378, A348694.

den[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]]; seq[m_] := Module[{s = Table[0, {m}], c = 0, n = 1, i}, While[c < m, i = den[n]; If[i <= m && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[100]

A348696 Numbers m such that there is at least one smaller number k < m with the same harmonic mean of divisors as m.

Original entry on oeis.org

135, 224, 496, 936, 1485, 1488, 1755, 2295, 2464, 2565, 2912, 3105, 3808, 3915, 4185, 4256, 4464, 4680, 4995, 5152, 5456, 5535, 5805, 6345, 6448, 6496, 6552, 6860, 6944, 7155, 7965, 8235, 8288, 8432, 9045, 9184, 9424, 9585, 9632, 9855, 10296, 10528, 10665, 10976

Offset: 1

Amiram Eldar, Oct 30 2021

nonn

The corresponding values of k (the least in case there are more than one) are 84, 120, 140, 864, 924, 420, 1092, 1428, 1320, 1596, ... (see the link for more values).

The least term m with more than one smaller number k with the same harmonic mean of divisors as m is m = a(1237) = A348697(1) = 321048 with k = 201096 and 296352.

			135 is a term since the harmonic mean of divisors of 135 is 9/2, and it is also the harmonic mean of divisors of 84 which is smaller than 135.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Amiram Eldar, Table of n, a(n), least k for n=1..1000

Cf. A099377, A099378, A348697.

h = Table[DivisorSigma[0, n]/DivisorSigma[-1, n], {n, 1, 10000}]; i = Position[(t = Tally[h])[[;; , 2]], _?(# > 1 &)] // Flatten; Position[h, #][[2 ;; -1]] & /@ t[[i, 1]] // Flatten // Sort

A348697 Numbers m such that there are at least two smaller numbers k < m with the same harmonic mean of divisors as m.

Original entry on oeis.org

321048, 448335, 1284192, 1605240, 1672125, 1862190, 3531528, 5016375, 5457816, 6420960, 7384104, 7621695, 8026200, 9310392, 9952488, 10311705, 11878776, 13001715, 13035330, 13162968, 13805064, 13898385, 14126112, 15089256, 16588395, 17015544, 17657640, 17836000

Offset: 1

Amiram Eldar, Oct 30 2021

nonn

What is the least term m with more than two smaller numbers k with the same harmonic mean of divisors as m?

The first such m is 44474832, which has harmonic mean of divisors 729/14, as do 5214132, 12553380 and 25676352. - Robert Israel, May 18 2025

			321048 is a term since the harmonic mean of divisors of 321048 is 3528/125, and it is also the harmonic mean of divisors of both 201096 and 296352 which are smaller than 321048.

Robert Israel, Table of n, a(n) for n = 1..101

Cf. A099377, A099378, A348696.

hmd:= proc(n) local d,D;
  D:= numtheory:-divisors(n);
  nops(D)/add(1/d, d = D)
end proc:
R:= NULL: count:= 0:
for m from 1 while count < 50 do
  v:= hmd(m);
  if assigned(C[v]) then
    C[v]:= C[v]+1;
    if C[v] >= 3 then
      R:= R,m; count:= count+1;
    fi
  else C[v]:= 1
  fi;
od:
R; # Robert Israel, May 18 2025

h = Table[DivisorSigma[0, n]/DivisorSigma[-1, n], {n, 1, 2*10^6}]; i = Position[(t = Tally[h])[[;; , 2]], _?(# > 2 &)] // Flatten; Position[h, #][[3 ;; -1]] & /@ t[[i, 1]] // Flatten // Sort

A348826 Numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

Original entry on oeis.org

36, 100, 144, 324, 400, 576, 900, 1296, 1600, 1936, 2304, 2500, 3600, 4356, 4624, 5184, 6400, 8100, 8464, 9216, 10000, 10404, 11664, 12100, 13456, 14400, 17424, 18496, 19044, 20736, 22500, 25600, 26244, 28900, 30276, 30976, 32400, 38416, 40000, 41616, 46656, 48400

Offset: 1

Amiram Eldar, Nov 01 2021

nonn

Since the harmonic mean of the divisors of k is k*tau(k)/sigma(k), where tau(k) = A000005(k) and sigma(k) = A000203(k), then A099378(k) <= sigma(k). Therefore, all the terms k have sigma(k) > 2*k and are thus abundant numbers (A005101).

The first nonsquare term is a(92) = 320000. Apparently, the nonsquares are relatively rare in this sequence. For example, of the first 10^4 terms only 107 are nonsquares.

			36 is a term since the harmonic mean of the divisors of 36 is 324/91 and 91 > 2*36 = 72.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A099377, A099378.
Subsequence of A005101 and A348825.
A348827 is a subsequence.

q[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > 2*n; Select[Range[50000], q]

isok(k) = my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > 2*k); \\ Michel Marcus, Nov 01 2021

A348827 Nonsquare numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

Original entry on oeis.org

320000, 941192, 1229312, 3001250, 5120000, 8000000, 14172488, 14623232, 15059072, 19668992, 35701250, 38614472, 42762752, 60236288, 66724352, 75031250, 121726800, 128000000, 143278592, 147061250, 168480000, 222814800, 226759808, 233971712, 257875200, 319813200

Offset: 1

Amiram Eldar, Nov 01 2021

nonn

The smallest term that is not twice a square is a(17) = 121726800 = 13 * 3060^2.

			320000 = 2 * 400^2 is a term since it is not a square, the harmonic mean of the divisors of 320000 is 16000000/798963 and 798963 > 2*320000 = 640000.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A001105, A099377, A099378.
Intersection of A000037 and A348826.
Subsequence of A005101 and A348825.

q[n_] := !IntegerQ @ Sqrt[n] && Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]] > 2*n; Select[Range[8000000], q]

isok(k) = if (!issquare(k), my(d=divisors(k)); (denominator(#d/sum(i=1, #d, 1/d[i])) > 2*k)); \\ Michel Marcus, Nov 01 2021

list(lim)=my(v=List()); forfactored(n=320000, lim\1, if(gcd(n[2][, 2])%2 && denominator(sigma(n, 0)/sigma(n, -1))>2*n[1], listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 01 2021

A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

Original entry on oeis.org

1, 2, 3, 6, 28, 40, 84, 120, 135, 224, 270, 672, 819, 1638, 3780, 10880, 13392, 30240, 32640, 32760, 167400, 950976, 1303533, 2178540, 2607066, 3138345, 4713984, 6276690, 8910720, 14705145, 17428320, 29410290, 45532800, 52141320, 179734464, 301953024, 311323824

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 3-smooth number. Of the 937 harmonic numbers below 10^14, 38 are terms in this sequence.
If a term is not a harmonic number, then its numerator and denominator of the harmonic mean of its divisors are powers of 2 and 3, or vice versa.
If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.

			2 is a term since the harmonic mean of its divisors is 4/3 = 2^2/3.
3 is a term since the harmonic mean of its divisors is 3/2.
40 is a term since the harmonic mean of its divisors is 32/9 = 2^5/3^2.

Amiram Eldar, Table of n, a(n) for n = 1..47

Cf. A000079, A000244, A001599, A001600, A003586, A099377, A099378.
Subsequence of A348868.
Similar sequences: A074266, A122254, A348658, A348659.

smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[10^5], q]

A348868 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 5-smooth numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 15, 24, 27, 28, 30, 40, 54, 84, 120, 135, 140, 216, 224, 270, 420, 496, 672, 756, 775, 819, 1080, 1120, 1488, 1550, 1638, 2176, 2325, 2480, 3360, 3780, 4095, 4650, 6048, 6200, 6528, 6552, 7440, 8190, 10880, 11375, 13392, 18600, 20925, 21700

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

The terms that are also harmonic numbers (A001599) are those whose harmonic mean of divisors (A001600) is a 5-smooth number. Of the 937 harmonic numbers below 10^14, 83 are terms in this sequence.

If k1 and k2 are coprime terms, then k1*k2 is also a term. In particular, if k is an odd term, then 2*k is also a term.

			8 is a term since the harmonic mean of its divisors is 32/15 and both 32 = 2^5 and 15 = 3*5 are 5-smooth numbers.

Amiram Eldar, Table of n, a(n) for n = 1..234

Cf. A001599, A001600, A051037, A099377, A099378.
A348867 is a subsequence.
Similar sequences: A074266, A348658, A348659.

smQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3] * 5^IntegerExponent[n, 5]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := smQ[Numerator[(hn = h[n])]] && smQ[Denominator[hn]]; Select[Range[22000], q]

A349497 a(n) is the smallest element in the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Nov 20 2021

nonn

			a(2) = 1 since the continued fraction of the harmonic mean of the divisors of 2, 4/3 = 1 + 1/3, has 2 elements, {1, 3}, and the smallest of them is 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001248, A001599, A001600, A099377, A099378, A129521, A349473.

a[n_] := Min[ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]]; Array[a, 100]

a(p) = 1 for a prime p.
a(p^2) = 1 for a prime p != 3.
a(A129521(n)) = 1 for n > 3.
For a harmonic number m = A001599(k), a(m) = A099377(m) = A001600(k).

A353345 Numbers k such that the elements of the continued fractions of the harmonic means of the divisors of k and k+1 are anagrams of each other.

Original entry on oeis.org

688126, 29900656, 35217656, 71624168, 154979487, 527560886, 871173148, 1370592266, 2461226804, 3232529461, 3232684430, 3431178214, 3471121856, 3486231973, 3527029430, 5732671200, 6258062402, 8784477355, 9334188311, 12670993089, 12707869077, 15120804392, 16317131894

Offset: 1

Amiram Eldar, Apr 15 2022

nonn

			688126 is a term since sequences of elements of the continued fractions of the harmonic means of the divisors of 688126 and 688127, 688126/70281 and 688127/77304, are {9, 1, 3, 1, 3, 1, 2, 9, 1, 1, 6, 8} and {8, 1, 9, 6, 3, 1, 2, 1, 3, 1, 1, 9} respectively, and they are anagrams of each other.

Cf. A069567, A099377, A099378, A349473, A353346, A353347.

Subsequence of A349499.

h[n_] := Sort[ContinuedFraction[DivisorSigma[0, n]/DivisorSigma[-1, n]]]; seq[max_] := Module[{s = {}, n = 2, c = 0, h1 = h[1], h2}, While[n < max, h2 = h[n]; If[h1 == h2, AppendTo[s, n - 1]]; h1 = h2; n++]; s]; seq[4*10^7]

A353691 a(n) is the least number k > n such that h(k)/h(n) is an integer, where h(n) is the harmonic mean of the divisors of n, or -1 if no such k exists.

Original entry on oeis.org

6, 120, 28, 234, 30, 270, 42, 29792, 252, 1120, 66, 234, 78, 840, 140, 200, 102, 2016, 114, 1170, 945, 1320, 138, 1080, 150, 1560, 756, 270, 174, 3360, 186, 1272960, 308, 2040, 210, 9720, 222, 2280, 364, 148960, 246, 1890, 258, 2574, 1260, 2760, 282, 600, 294

Offset: 1

Amiram Eldar, May 04 2022

nonn

Does a(n) exist for all n? If m is a harmonic number (A001599) and gcd(n, m) = 1, then a(n) exists and a(n) <= m*n, since h(m*n) = h(m)*h(n) and h(m) is an integer.

			a(2) = 120 since 120 is the least number > 2 such that h(120)/h(2) = (16/3)/(4/3) = 4 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..639

Cf. A001599, A099377, A099378.

Similar sequences: A069789, A069797, A069805, A353692.

h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; a[n_] := Module[{k = n + 1, hn = h[n]}, While[!IntegerQ[h[k]/hn], k++]; k]; Array[a, 30]

from math import prod, gcd
from sympy import factorint
def A353691_helper(n):
    f = factorint(n).items()
    return prod(p**e*(p-1)*(e+1) for p, e in f), prod(p**(e+1)-1 for p, e in f)
def A353691(n):
    Hnp, Hnq = A353691_helper(n)
    g = gcd(Hnp, Hnq)
    Hnp //= g
    Hnq //= g
    k = n+1
    Hkp, Hkq = A353691_helper(k)
    while (Hkp*Hnq) % (Hkq*Hnp):
        k += 1
        Hkp, Hkq = A353691_helper(k)
    return k # Chai Wah Wu, May 07 2022

a(p) = 6*p for a prime p > 3.

cp's OEIS Frontend

A348695 a(n) is the least number k such that the denominator of the harmonic mean of the divisors of k is equal to n, or -1 if no such k exists.

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A348696 Numbers m such that there is at least one smaller number k < m with the same harmonic mean of divisors as m.

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A348697 Numbers m such that there are at least two smaller numbers k < m with the same harmonic mean of divisors as m.

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A348826 Numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

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A348827 Nonsquare numbers k such that the denominator of the harmonic mean of the divisors of k is larger than 2*k.

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A348867 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 3-smooth numbers.

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A348868 Numbers whose numerator and denominator of the harmonic mean of their divisors are both 5-smooth numbers.

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A349497 a(n) is the smallest element in the continued fraction of the harmonic mean of the divisors of n.

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