A099378 -id:A099378 - cp's OEIS frontend

A179971 Positions of records in the sequence of harmonic means, i.e., in the sequence of rationals A099377(.)/A099378(.).

1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 108, 120, 144, 168, 180, 240, 336, 360, 420, 480, 504, 630, 720, 840, 1008, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 3780, 3960, 4200, 4320, 4620, 5040, 7560, 9240, 10080, 12600, 13860, 15120

Offset: 1

Robert G. Wilson v, Aug 04 2010

nonn

			The sequence of harmonic means starts 1 < 4/3 < 3/2 < 12/7, increasing from the first to the fourth, which adds 1 to 4 to the sequence.
The fifth harmonic mean is 5/3, smaller than 12/7 and not a record, so 5 is not in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Ore's Conjecture

Cf. A099377, A099378.

hm := proc(n) option remember; n* numtheory[tau](n)/numtheory[sigma](n) ; end proc:
A179971 := proc(n) option remember; if n = 1 then 1; else for k from procname(n-1)+1 do if hm(k) > hm(procname(n-1)) then return k; end if; end do; end if; end proc:
seq(A179971(n),n=1..40) ; # R. J. Mathar, Aug 06 2010

f[n_] := f[n] = DivisorSigma[0, n]/Plus @@ (1/Divisors@n); k = 1; mx = 0; lst = {}; While[k < 18480, a = f@k; If[a > mx, mx = a; AppendTo[lst, k]]; k++ ]; lst

Definition rephrased by R. J. Mathar, Aug 06 2010

A348865 Numbers k such that A099378(k) | (A099377(k) - 1).

Original entry on oeis.org

1, 2, 3, 6, 9, 14, 15, 24, 28, 30, 32, 42, 44, 55, 84, 91, 95, 120, 135, 140, 182, 198, 224, 234, 243, 261, 270, 308, 330, 351, 420, 444, 459, 477, 483, 492, 496, 546, 564, 570, 625, 630, 636, 672, 744, 756, 840, 852, 861, 924

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

A disjoint union of the harmonic numbers (A001599) and the numbers whose harmonic mean of divisors is of the form m + 1/k, where m and k are integers.

			2 is a term since the harmonic mean of divisors of 2 is 4/3 and 3 | (4-1).

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

Cf. A001599, A099377, A099378, A348866.

h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := Divisible[Numerator[(h1 = h[n])] - 1, Denominator[h1]]; Select[Range[1000], q]

A348866 Composite numbers k such that A099378(k) | (A099377(k) + 1).

Original entry on oeis.org

6, 15, 20, 28, 33, 35, 42, 51, 66, 69, 70, 84, 87, 114, 117, 123, 135, 138, 140, 141, 153, 159, 177, 186, 204, 207, 210, 213, 249, 258, 267, 270, 273, 276, 282, 285, 297, 303, 308, 321, 339, 348, 354, 357, 372, 393, 399, 402, 411, 420, 426, 432, 435, 447, 464

Offset: 1

Amiram Eldar, Nov 02 2021

nonn

A disjoint union of the harmonic numbers (A001599) and the composite numbers whose harmonic mean of divisors is of the form m - 1/k, where m and k are integers.
If p is an odd prime, then the harmonic mean of its divisors is p*tau(p)/sigma(p) = p*A000005(p)/A000203(p) = p/((p+1)/2), so A099378(p) | (A099377(p) + 1). Therefore, this sequence is restricted to composite numbers.
This sequence is infinite. For example, it includes all the semiprimes of the form 3*p, where p == 2 (mod 3).

			15 is a term since it is composite, the harmonic mean of divisors of 15 is 5/2 and 2 | (5+1).

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

Cf. A000005, A000203, A001599, A003627, A099377, A099378, A348865.

h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := Divisible[Numerator[(h1 = h[n])] + 1, Denominator[h1]]; Select[Range[1000], CompositeQ[#] && q[#] &]

A348847 Numbers k where the ratios A099378(k)/k reach a record value.

Original entry on oeis.org

1, 2, 4, 8, 16, 36, 144, 576, 900, 3600, 14400, 32400, 129600, 435600, 6969600, 8643600, 34574400, 77792400, 311169600, 2498000400, 2800526400, 7779240000, 9992001600, 22482003600

Offset: 1

Amiram Eldar, Nov 02 2021

Are there nonsquare terms in this sequence that are larger than 8?

			The first 4 ratios A099378(k)/k, for k = 1 to 4, are 1, 3/2, 2/3 and 7/4. The record values occur at k = 1, 2 and 4, the first 3 terms of this sequence.

Cf. A099377, A099378, A348414.

r[n_] := Denominator[DivisorSigma[0, n]/DivisorSigma[-1, n]]/n; s = {}; rm = 0; Do[If[(r1 = r[n]) > rm, rm = r1; AppendTo[s, n]], {n, 1, 5*10^5}]; s

A099377 Numerators of the harmonic means of the divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 12, 5, 2, 7, 32, 27, 20, 11, 18, 13, 7, 5, 80, 17, 36, 19, 20, 21, 22, 23, 16, 75, 52, 27, 3, 29, 10, 31, 64, 11, 68, 35, 324, 37, 38, 39, 32, 41, 7, 43, 22, 45, 23, 47, 120, 49, 100, 17, 156, 53, 18, 55, 56, 57, 116, 59, 30, 61, 31, 189, 448, 65, 11, 67, 68, 23, 35

Offset: 1

Eric W. Weisstein, Oct 13 2004

			1, 4/3, 3/2, 12/7, 5/3, 2, 7/4, 32/15, ...

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ore's Conjecture

Cf. A099378.

f[n_] := DivisorSigma[0, n]/Plus @@ (1/Divisors@n); Numerator@ Array[f, 70] (* Robert G. Wilson v, Aug 04 2010 *)
Table[Numerator[DivisorSigma[0, n]/DivisorSigma[-1, n]], {n, 70}] (* Ivan Neretin, Nov 13 2016 *)

a(n) = my(d=divisors(n)); numerator(#d/sum(k=1, #d, 1/d[k])); \\ Michel Marcus, Nov 13 2016

from sympy import gcd, divisor_sigma
def A099377(n): return (lambda x, y: y*n//gcd(x,y*n))(divisor_sigma(n),divisor_sigma(n,0)) # Chai Wah Wu, Oct 20 2021

More terms from Robert G. Wilson v, Aug 04 2010

A349473 Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 1, 3, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 3, 2, 7, 2, 2, 13, 2, 4, 2, 1, 1, 5, 2, 1, 1, 3, 1, 1, 6, 2, 3, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 8, 2, 1, 3, 3, 1, 1, 9, 2, 1, 6, 2, 1, 1, 1, 2, 2, 2, 4, 1, 1, 11, 3, 5, 2, 2, 2, 1, 1, 2, 2, 2, 10, 2, 1, 2, 3, 3, 1, 1, 14

Offset: 1

Amiram Eldar, Nov 19 2021

For an odd prime p > 3, the p-th row has a length 3 with a(p, 1) = a(p, 2) = 1 and a(p, 3) = (p-1)/2.

For a harmonic number m = A001599(k), the m-th row has a length 1 with a(k, 1) = A099377(m) = A001600(k).

			The first ten rows of the triangle are:
  1,
  1, 3,
  1, 2,
  1, 1, 2, 2,
  1, 1, 2,
  2,
  1, 1, 3,
  2, 7, 2,
  2, 13,
  2, 4, 2
  ...

Cf. A001599, A001600, A099377, A099378.

Cf. A349474 (row lengths).

row[n_] := ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; Table[row[k], {k, 1, 29}] // Flatten

A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 16, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 8, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 80, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 28, 57, 116, 59, 4, 61, 31, 63, 128, 65, 11, 67, 136, 23, 35, 71, 16, 73

Offset: 1

Emeric Deutsch, Jan 31 2005

			1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 16 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A103340 (denominators), A099377, A099378.

Cf. A005117, A034444, A034448, A077610.

import Data.Ratio ((%), numerator)
a103339 = numerator . uhm where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

with(numtheory): udivisors:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k],n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: utau:=n->nops(udivisors(n)): usigma:=n->sum(udivisors(n)[j],j=1..nops(udivisors(n))): uH:=n->n*utau(n)/usigma(n):seq(numer(uH(n)),n=1..81);

ud[n_] := 2^PrimeNu[n]; usigma[n_] := DivisorSum[n, If[GCD[#, n/#] == 1, #, 0]&]; a[1] = 1; a[n_] := Numerator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *)
a[n_] := Numerator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 10 2023 *)

a(n) = {my(f = factor(n)); numerator(n * prod(i=1, #f~, 2/(1 + f[i, 1]^f[i, 2]))); } \\ Amiram Eldar, Mar 10 2023

from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103339(n): return (lambda x, y: y*n//gcd(x,y*n))(udivisor_sigma(n),udivisor_sigma(n,0)) # Chai Wah Wu, Oct 20 2021

a(A006086(n)) = A006087(n). - Reinhard Zumkeller, Mar 17 2012
From Amiram Eldar, Mar 10 2023: (Start)
a(n)/A103340(n) = n*A034444(n)/A034448(n).
a(n)/A103340(n) <= A099377(n)/A099378(n), with equality if and only if n is squarefree (A005117). (End)

A103340 Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

Original entry on oeis.org

1, 3, 2, 5, 3, 1, 4, 9, 5, 9, 6, 5, 7, 3, 2, 17, 9, 5, 10, 3, 8, 9, 12, 3, 13, 21, 14, 5, 15, 3, 16, 33, 4, 27, 12, 25, 19, 15, 14, 27, 21, 2, 22, 15, 1, 9, 24, 17, 25, 39, 6, 35, 27, 7, 18, 9, 20, 45, 30, 1, 31, 12, 20, 65, 21, 3, 34, 45, 8, 9, 36, 5, 37, 57, 26, 25, 24, 7, 40, 51, 41, 63

Offset: 1

Emeric Deutsch, Jan 31 2005

			1, 4/3, 3/2, 8/5, 5/3, 2, ...
a(8) = 9 because the unitary divisors of 8 are {1,8} and 2/(1/1 + 1/8) = 16/9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A103339 (numerators), A099377, A099378.

Cf. A034444, A034448, A077610.

import Data.Ratio ((%), denominator)
a103340 = denominator . uhm where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

with(numtheory): udivisors:=proc(n) local A, k: A:={}: for k from 1 to tau(n) do if gcd(divisors(n)[k], n/divisors(n)[k])=1 then A:=A union {divisors(n)[k]} else A:=A fi od end: utau:=n->nops(udivisors(n)): usigma:=n->sum(udivisors(n)[j],j=1..nops(udivisors(n))): uH:=n->n*utau(n)/usigma(n):seq(denom(uH(n)),n=1..90);

ud[n_] := 2^PrimeNu[n]; usigma[n_] := DivisorSum[n, If[GCD[#, n/#] == 1, #, 0]&]; a[1] = 1; a[n_] := Denominator[n*ud[n]/usigma[n]]; Array[a, 100] (* Jean-François Alcover, Dec 03 2016 *)
a[n_] := Denominator[n * Times @@ (2 / (1 + Power @@@ FactorInteger[n]))]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Mar 10 2023 *)

a(n) = {my(f = factor(n)); denominator(n * prod(i=1, #f~, 2/(1 + f[i, 1]^f[i, 2]))); } \\ Amiram Eldar, Mar 10 2023

from sympy import gcd
from sympy.ntheory.factor_ import udivisor_sigma
def A103340(n): return (lambda x, y: x//gcd(x,y*n))(udivisor_sigma(n),udivisor_sigma(n,0)) # Chai Wah Wu, Oct 20 2021

a(A006086(n)) = 1. - Reinhard Zumkeller, Mar 17 2012

A349474 a(n) is the length of the continued fraction of the harmonic mean of the divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 3, 1, 3, 3, 2, 3, 3, 4, 3, 2, 2, 7, 3, 4, 3, 3, 5, 3, 3, 2, 6, 3, 4, 1, 3, 2, 3, 2, 3, 4, 3, 8, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 5, 6, 4, 3, 4, 3, 4, 2, 5, 5, 7, 3, 3, 3, 5, 7, 7, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 7, 4, 4, 4, 3, 4, 6, 6, 3, 2, 4, 6, 3

Offset: 1

Amiram Eldar, Nov 19 2021

nonn

a(n) = 1 if and only if n is a harmonic number (A001599).

a(n) <= 2 if and only if n is in A348865.

			a(1) = 1 since the harmonic mean of the divisors of 1 is 1 and its continued fraction has 1 element, {1}.
a(2) = 2 since the harmonic mean of the divisors of 2 is 4/3 = 1 + 1/3 and its continued fraction has 2 elements, {1, 3}.
a(4) = 4 since the harmonic mean of the divisors of 4 is 12/7 = 1 + 1/(1 + 1/(2 + 1/2)) and its continued fraction has 4 elements, {1, 1, 2, 2}.

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

Row length of A349473.

Cf. A001599, A071862, A099377, A099378, A348865.

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

A074266 Numbers k such that the harmonic mean of the divisors of k is the square of a rational number.

Original entry on oeis.org

1, 216, 468, 810, 1550, 1638, 3744, 10880, 11340, 13965, 21700, 23716, 40176, 45847, 50274, 56896, 80262, 90720, 97969, 126360, 128744, 137940, 139159, 161728, 173600, 189728, 224450, 319579, 434511, 482790, 515450, 526500, 555660

Offset: 1

Joseph L. Pe, Sep 20 2002

nonn

			The harmonic mean of the divisors of 468 is 324/49 = (18/7)^2, the square of a rational number, so 468 is a term of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..500

Cf. A001599, A099377, A099378.

H[l_] := Module[{m, s}, m = Length[l]; s = 0; For[i = 1, i <= m, i++, s = s + (1/l[[i]])]; s = s/m; s = 1/s; s] r = {}; Do[d = Divisors[n]; h = H[d]; num = Numerator[h]; den = Denominator[h]; If[IntegerQ[num^(1/2)] && IntegerQ[den^(1/2)], r = Append[r, n]], {n, 1, 10^6}]; r

cp's OEIS Frontend

A179971 Positions of records in the sequence of harmonic means, i.e., in the sequence of rationals A099377(.)/A099378(.).

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A348865 Numbers k such that A099378(k) | (A099377(k) - 1).

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A348866 Composite numbers k such that A099378(k) | (A099377(k) + 1).

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Mathematica

A348847 Numbers k where the ratios A099378(k)/k reach a record value.

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Mathematica

A099377 Numerators of the harmonic means of the divisors of the positive integers.

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A349473 Irregular triangle read by rows: the n-th row contains the elements in the continued fraction of the harmonic mean of the divisors of n.

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Mathematica

A103339 Numerator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

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A103340 Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.

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