A001600 -id:A001600 - cp's OEIS frontend

A335316 Harmonic numbers (A001599) with a record harmonic mean of divisors.

1, 6, 28, 140, 270, 672, 1638, 2970, 8190, 27846, 30240, 167400, 237510, 332640, 695520, 1421280, 2178540, 2457000, 11981970, 14303520, 17428320, 23963940, 27027000, 46683000, 56511000, 71253000, 142990848, 163390500, 164989440, 191711520, 400851360, 407386980

Offset: 1

Amiram Eldar, May 31 2020

nonn

The corresponding record values are 1, 2, 3, 5, 6, 8, 9, 11, 15, ... (see the link for more values).

The terms 1, 6, 30240 and 332640 are also terms of A179971.

			The first 7 harmonic numbers are 1, 6, 28, 140, 270, 496 and 672. Their harmonic means of divisors (A001600) are 1, 2, 3, 5, 6, 5 and 8. The record values, 1, 2, 3, 5, 6 and 8 occur at 1, 6, 28, 140, 270 and 672, the first 6 terms of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..70 (terms below 10^14)
Amiram Eldar, Table of n, a(n), A099377(a(n)) for n = 1..70

Cf. A001599, A001600, A099377, A099378, A179971, A335317, A335318.

h[n_] := n * DivisorSigma[0, n] / DivisorSigma[1, n]; hm = 0; s = {}; Do[h1 = h[n];  If[IntegerQ[h1] && h1 > hm, hm = h1; AppendTo[s, n]], {n, 1, 10^6}]; s

A157849 Numbers k such that are not harmonic means of divisors of harmonic (Ore) numbers (harmonic (Ore) numbers is A001599).

Original entry on oeis.org

4, 12, 16, 18, 20, 22, 23, 28, 30, 32, 33, 34, 36, 38, 40, 43, 52, 55, 56, 57, 58, 59, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 74, 76, 79, 90, 93, 95, 98, 100, 103, 104, 109, 111, 112, 113, 119, 122, 123, 124, 126, 129, 131, 133, 134, 136, 137, 138, 141, 142, 146, 148, 151, 154, 157, 162, 170

Offset: 1

Jaroslav Krizek, Mar 07 2009

nonn

a(n) non-occurring in A001600(m) = A001599(m)*tau(A001599(m))/sigma(A001599(m)) = A001599(m)*A000005(A001599(m))/A000203(A001599(m)).

Robert G. Wilson v, Table of n, a(n) for n = 1..412
Takeshi Goto, On Ore's harmonic numbers, slides, Tokyo University of Science, Japan.

Cf. A001600, A001599, A000005, A000203.

More terms from Robert G. Wilson v, Aug 18 2013

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).

A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 11, 13, 13, 10, 7, 15, 16, 15, 9, 20, 18, 14, 25, 24, 19, 25, 15, 27, 28, 30, 18, 36, 13, 21, 17, 29, 40, 33, 24, 28, 38, 31, 29, 45, 34, 27, 28, 44, 27, 60, 36, 52, 46, 26, 51, 42, 55, 33, 66, 40, 24, 37, 49, 29, 47, 57, 34, 68, 49, 44

Offset: 1

Amiram Eldar, Mar 09 2023

nonn

Each term appears a finite number of times in the sequence (Hagis and Cohen, 1990).

Amiram Eldar, Table of n, a(n) for n = 1..239
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A063947, A361316, A361317.

Similar sequences: A001600, A006087.

f[p_, e_] := Module[{b = IntegerDigits[e, 2], m}, m = Length[b]; Product[If[b[[j]] > 0, 2/(1 + p^(2^(m - j))), 1], {j, 1, m}]]; s[1] = 1; s[n_] := n * Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^4], IntegerQ]

ihmean(n) = {my(f = factor(n), b); n * prod(i=1, #f~, b = binary(f[i, 2]); prod(k=1, #b, if(b[k], 2/(f[i, 1]^(2^(#b-k))+1), 1))); };
lista(kmax) = {my(ih); for(k = 1, kmax, ih = ihmean(k); if(denominator(ih) == 1, print1(ih, ", ")));}

a(n) = A361316(A063947(n)).

A361784 Harmonic means the bi-unitary divisors of the bi-unitary harmonic numbers (A286325).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 7, 7, 8, 11, 13, 13, 12, 10, 16, 7, 18, 16, 15, 24, 15, 20, 20, 18, 14, 22, 25, 24, 19, 25, 23, 27, 33, 31, 44, 32, 34, 30, 25, 36, 13, 46, 31, 21, 29, 40, 38, 33, 28, 40, 48, 38, 29, 45, 34, 47, 28, 32, 32, 44, 60, 27, 32, 28, 46, 26, 51

Offset: 1

Amiram Eldar, Mar 24 2023

nonn

			a(3) = 3 since A286325(3) = 45, the bi-unitary divisors of 45 are 1, 5, 9, and 45, and their harmonic mean is 3.

Amiram Eldar, Table of n, a(n) for n = 1..313
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A286324, A286325, A361782, A361783.

Similar sequences: A001600, A006087, A361318.

f[p_, e_] := p^e * If[OddQ[e], (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2))]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 10^5], IntegerQ]

bhmean(n) = {my(f = factor(n), p, e); n * prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2];  if(e%2, (e + 1)*(p - 1)/(p^(e + 1) - 1), e/((p^(e + 1) - 1)/(p - 1) - p^(e/2)))); }
lista(kmax) = {my(bh); for(k = 1, kmax, bh = bhmean(k); if(denominator(bh) == 1, print1(bh, ", "))); }

a(n) = A361782(A286325(n)).

A376713 a(n) is the minimum number with exactly n triples (u,m,v) of divisors such that u < v and m^2 is the harmonic mean of u^2 and v^2.

Original entry on oeis.org

35, 70, 140, 210, 560, 420, 2240, 840, 1260, 1680, 35840, 2100, 143360, 6720, 5040, 4200, 1709435, 6300, 645575, 8400, 20160, 77350, 36728125, 12600, 45360, 430080, 31500, 33600, 1117484375, 25200, 24171875, 29400, 154700, 3418870, 181440, 44100, 31633175, 1291150

Offset: 1

Michel Lagneau, Oct 02 2024

For two integers u, v, by definition the harmonic mean m is given by 2/m = 1/u + 1/v.
We observe that a(n) is divisible by 35.
Using the observed (checked through n=200) relation a(n) <= 35*A005179(n), further terms from a(29) to a(52) are (<=9395240960, 25200, <=37580963840, 29400, 154700, 3418870, 181440, 44100, <=2405181685760, 1291150, 1290240, 58800, <=38482906972160, 100800, <=153931627888640, 232050, 126000, <=440401920, <=2462906046218240, 88200, 1632960, 226800, 6837740, 2150400). - Hugo Pfoertner, Oct 05 2024

			a(1) = 35 is because the D2(35) = {1, 5^2, 7^2, 35^2} with the unique pair of squares of divisors (5^2, 35^2) we obtain 1/5^2+1/35^2 = 2/7^2. Hence m = 7^2 is in D2(35). There is no other solution.
a(2) = 70 because D2(70) = {1, 2^2, 5^2, 7^2, 10^2, 14^2, 35^2, 70^2} and we find two pairs of squares of divisors: (5^2, 35^2) and (10^2, 70^2) giving respectively:
First solution: 1/5^2+1/35^2 = 2/7^2. Hence m = 7 ^2 is in D2(70);
Second solution: 1/10^2+1/70^2 = 2/14^2. Hence m = 14^2 is in D2(70).

Lucas A. Brown, Python program.

Cf. A001599, A001600, A005179.

with(numtheory):nn:=5*10^8:
for n from 1 to 25 do:
 ii:=0:
for k from 4 to nn while (ii=0)do:
d:=divisors(k):n0:=nops(d):it:=0:
  for i from 1 to n0-1 do:
   for j from i+1 to n0 do:
     s:=1/d[i]^2+ 1/d[j]^2:
      for u from 1 to n0 do:
       if s=2/d[u]^2 then it:=it+1:else
       fi:
      od:
    od:
   od:
   if it=n then ii:=1:printf(`%d %d \n`,n,k):else fi:
od:od:

a(26)-a(28) from Hugo Pfoertner, Oct 03 2024

a(17), a(19), a(23), and a(29)-a(38) from Lucas A. Brown, Nov 04 2024

cp's OEIS Frontend

A335316 Harmonic numbers (A001599) with a record harmonic mean of divisors.

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A157849 Numbers k such that are not harmonic means of divisors of harmonic (Ore) numbers (harmonic (Ore) numbers is A001599).

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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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A361318 Harmonic means of the infinitary divisors of the infinitary harmonic numbers.

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A361784 Harmonic means the bi-unitary divisors of the bi-unitary harmonic numbers (A286325).

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A376713 a(n) is the minimum number with exactly n triples (u,m,v) of divisors such that u < v and m^2 is the harmonic mean of u^2 and v^2.

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