A099377 -id:A099377 - cp's OEIS frontend

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

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

A250094 Positive integers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

Original entry on oeis.org

1, 3, 5, 7, 11, 13, 17, 19, 20, 21, 22, 23, 27, 29, 31, 35, 37, 38, 39, 41, 43, 45, 47, 49, 53, 55, 56, 57, 59, 61, 65, 67, 68, 71, 73, 77, 79, 83, 85, 86, 89, 93, 97, 99, 101, 103, 107, 109, 110, 111, 113, 115, 116, 118, 119, 125, 127, 129, 131, 133, 134

Offset: 1

Colin Barker, Nov 12 2014

nonn

A subsequence of A099377: n such that A099377(n) = n.

All odd primes are in this sequence.

			20 is a term because the divisors of 20 are [1,2,4,5,10,20] and 6 / (1/1 + 1/2 + 1/4 + 1/5 + 1/10 + 1/20) = 20/7.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Colin Barker)

Cf. A099377, A247081, A250095.

Select[Range[200],Numerator[HarmonicMean[Divisors[#]]]==#&] (* Harvey P. Dale, May 24 2017 *)
Select[Range[134], Numerator[DivisorSigma[0, #] * #/DivisorSigma[1, #]] == # &] (* Amiram Eldar, Mar 02 2020 *)

harmonicmean(v) = #v / sum(k=1, #v, 1/v[k])
s=[]; for(n=1, 500, if(numerator(harmonicmean(divisors(n)))==n, s=concat(s, n))); s

A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 32, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 128, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 48, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 256, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 09 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Hagis, Jr. and Graeme L. Cohen, Infinitary harmonic numbers, Bull. Australian Math. Soc., Vol. 41, No. 1 (1990), pp. 151-158.

Cf. A036537, A037445, A049417, A077609, A063947, A138302, A361317 (denominators).

Similar sequences: A099377, A103339.

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}]]; a[1] = 1; a[n_] := Numerator[n * Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), b); numerator(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)))); }

a(n) = numerator(n*A037445(n)/A049417(n)).
a(n)/A361317(n) <= A099377(n)/A099378(n), with equality if and only if n is in A036537.
a(n)/A361317(n) >= A103339(n)/A103340(n), with equality if and only if n is in A138302.

A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

Original entry on oeis.org

1, 4, 3, 8, 5, 2, 7, 32, 9, 20, 11, 12, 13, 7, 5, 64, 17, 12, 19, 8, 21, 22, 23, 16, 25, 52, 27, 14, 29, 10, 31, 64, 11, 68, 35, 72, 37, 38, 39, 32, 41, 7, 43, 44, 3, 23, 47, 32, 49, 100, 17, 104, 53, 18, 55, 56, 57, 116, 59, 4, 61, 31, 63, 384, 65, 11, 67, 136

Offset: 1

Amiram Eldar, Mar 24 2023

			Fractions begin with 1, 4/3, 3/2, 8/5, 5/3, 2, 7/4, 32/15, 9/5, 20/9, 11/6, 12/5, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jozsef Sandor, On bi-unitary harmonic numbers, arXiv:1105.0294 [math.NT], 2011.

Cf. A188999, A222266, A286324, A361783 (denominators).

Similar sequences: A099377, A103339, A361316.

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))]; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100]

a(n) = {my(f = factor(n), p, e); numerator(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))))); }

a(n) = numerator(n*A286324(n)/A188999(n)).

f(n) = a(n)/A361783(n) is multiplicative with f(p^e) = (e+1)*(p-1)/(p^(e+1)-1) if e is odd, and e/((p^(e+1)-1)/(p-1) - p^(e/2)) if e is even.

A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.

Original entry on oeis.org

1, 6, 140, 496, 672, 2970, 27846, 105664, 173600, 237510, 539400, 695520, 726180, 753480, 1421280, 1539720, 2229500, 2290260, 8872200, 11981970, 14303520, 15495480, 33550336, 50401728, 71253000, 80832960, 90409410, 144963000, 221557248, 233103780, 287425800, 318177800

Offset: 1

Amiram Eldar, Jun 03 2020

nonn

If k is a harmonic number (A001599) and p is a prime that does not divide k, then k*p is a harmonic number if and only if (p+1)/2 is a divisor of the harmonic mean of the divisors of k, h(k) = k*tau(k)/sigma(k) = k*A000005(k)/A000203(k). The terms of this sequence are harmonic numbers k such that for all the divisors d of h(k), 2*d - 1 is either a nonprime or a prime divisor of k.

The even perfect numbers, 2^(p-1)*(2^p - 1) where p is a Mersenne exponent (A000043), have harmonic mean of divisors p. Therefore, they are in this sequence if p = 2 or if 2*p - 1 is composite (i.e., not in A172461). Of the first 47 Mersenne exponents there are 37 such primes (p = 2, 5, 13, 17, ...), with the corresponding even perfect numbers 6, 496, 33550336, 8589869056, ...

			1 is a term since it is a harmonic number, and there is no prime p such that 1*p = p is a harmonic number (if p is a prime, h(p) = 2*p/(p+1) cannot be an integer).

Amiram Eldar, Table of n, a(n) for n = 1..246 (terms below 10^14)
Mariano Garcia, On numbers with integral harmonic mean, The American Mathematical Monthly, Vol. 61, No. 2 (1954), pp. 89-96. See page 95.

Cf. A000005, A000043, A000203, A000396, A001599, A099377, A099378, A172461, A335368, A335370, A335371.

harmNums = Cases[Import["https://oeis.org/A001599/b001599.txt", "Table"], {, }][[;; , 2]]; harMean[n_] := n * DivisorSigma[0, n]/DivisorSigma[1, n]; primeCountQ[n_] := Module[{d = Divisors[harMean[n]]}, Select[2*d - 1, PrimeQ[#] && ! Divisible[n, #] &] == {}]; Select[harmNums, primeCountQ]

A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 6, 15, 21, 28, 140, 182, 496, 546, 672, 918, 1890, 2016, 4005, 4590, 24384, 52780, 55860, 68200, 84812, 90090, 105664, 145782, 186992, 204600, 381654, 728910, 907680, 1655400, 2302344, 2862405, 3828009, 3926832, 5959440, 21059220, 33550336, 33839988, 42325920

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

Terms that also Fibonacci numbers are 1, 3, 5, 21, and no more below Fibonacci(300).

			3 is a term since the harmonic mean of its divisors is 3/2 = Fibonacci(4)/Fibonacci(3).
15 is a term since the harmonic mean of its divisors is 5/2 = Fibonacci(5)/Fibonacci(3).

Cf. A000045, A099377, A099378.

Similar sequences: A074266, A123193, A272412, A272440, A348659.

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[{5 n^2 - 4, 5 n^2 + 4}]; h[n_] := DivisorSigma[0, n]/DivisorSigma[-1, n]; q[n_] := fibQ[Numerator[(hn = h[n])]] && fibQ[Denominator[hn]]; Select[Range[1000], q]

from itertools import islice
from sympy import integer_nthroot, gcd, divisor_sigma
def A348658(): # generator of terms
    k = 1
    while True:
        a, b = divisor_sigma(k), divisor_sigma(k,0)*k
        c = gcd(a,b)
        n1, n2 = 5*(a//c)**2-4, 5*(b//c)**2-4
        if (integer_nthroot(n1,2)[1] or integer_nthroot(n1+8,2)[1]) and (integer_nthroot(n2,2)[1] or integer_nthroot(n2+8,2)[1]):
            yield k
        k += 1
A348658_list = list(islice(A348658(),10)) # Chai Wah Wu, Oct 28 2021

A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

Original entry on oeis.org

3, 5, 13, 14, 15, 37, 42, 61, 66, 73, 92, 114, 157, 182, 193, 258, 277, 308, 313, 397, 402, 421, 457, 476, 477, 541, 546, 570, 613, 661, 673, 733, 744, 757, 812, 877, 978, 997, 1093, 1148, 1153, 1201, 1213, 1237, 1266, 1278, 1321, 1381, 1428, 1453, 1621, 1657

Offset: 1

Amiram Eldar, Oct 28 2021

nonn

The prime terms of this sequence are the primes p such that (p+1)/2 is also a prime (A005383).

If p is in A109835, then p*(2*p-1) is a semiprime term.

			3 is a term since the harmonic mean of its divisors is 3/2 and both 2 and 3 are primes.

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

Cf. A005383, A099377, A099378, A109835.

Similar sequences: A023194, A048968, A074266, A348659.

q[n_] := Module[{h = DivisorSigma[0, n]/DivisorSigma[-1, n]}, And @@ PrimeQ[{Numerator[h], Denominator[h]}]]; Select[Range[2000], q]

A349476 Numbers k such that the continued fraction of the harmonic mean of the divisors of k contains a single distinct element.

Original entry on oeis.org

1, 6, 15, 28, 30, 140, 270, 496, 545, 672, 792, 1365, 1638, 2970, 3515, 6200, 8128, 8190, 18600, 18620, 27846, 30240, 32760, 44950, 46359, 55860, 59670, 105664, 117800, 167400, 173600, 237510, 242060, 253539, 332640, 360360, 539400, 681156, 691782, 695520, 726180, 753480, 950976

Offset: 1

Amiram Eldar, Nov 19 2021

nonn

All the harmonic numbers (A001599) are terms of this sequence.
The least term with m elements in the continued fraction of the harmonic mean of its divisors for m = 1, 2, 3, and 4 is 1, 15, 792 and 545, respectively.
Are there terms with more than 4 elements? There are no such terms below 2*10^9.

			15 is a term since the harmonic mean of its divisors is 5/2 = 2 + 1/2.
545 is a term since the harmonic mean of its divisors is 109/33 = 3 + 1/(3 + 1/(3 + 1/3)).
792 is a term since the harmonic mean of its divisors is 528/65 = 8 + 1/(8 + 1/8).

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A099377, A099378, A349473.

c[n_] := ContinuedFraction[DivisorSigma[0, n] / DivisorSigma[-1, n]]; q[n_] := Length[Union[c[n]]] == 1; Select[Range[10^6], q]

A379945 Irregular triangle read by rows: T(n, k) is the numerator of the harmonic mean of all positive divisors of n except the k-th of them.

Original entry on oeis.org

2, 1, 3, 1, 8, 8, 4, 5, 1, 3, 2, 9, 18, 7, 1, 24, 24, 24, 12, 9, 9, 3, 15, 30, 15, 30, 11, 1, 15, 30, 5, 12, 30, 20, 13, 1, 21, 42, 21, 42, 5, 45, 15, 45, 64, 64, 64, 64, 32, 17, 1, 30, 3, 30, 5, 90, 45, 19, 1, 50, 25, 100, 50, 5, 100, 63, 63, 63, 63, 33, 66, 33, 66, 23, 1

Offset: 2

Stefano Spezia, Jan 07 2025

			The irregular triangle begins as:
   2,  1;
   3,  1;
   8,  8,  4;
   5,  1;
   3,  2,  9, 18;
   7,  1;
  24, 24, 24, 12;
   9,  9,  3;
  15, 30, 15, 30;
  ...
The irregular triangle of the related fractions begins as:
     2,     1;
     3,     1;
   8/3,   8/5,   4/3;
     5,     1;
     3,     2,   9/5,  18/11;
   7,1;
  24/7, 24/11, 24/13,   12/7;
   9/2,   9/5,   3/2;
  15/4, 30/13,  15/8,  30/17;
  ...

Stefano Spezia, Table of n, a(n) for n = 2..10371 (first 1400 rows of the triangle)
Jaba Kalita and Helen K. Saikia, A note on near harmonic divisor number and associated concepts, Palestine Journal of Mathematics, Vol. 13(4), 2024.

Cf. A000005, A000203, A001599, A027750, A099377, A379946 (denominator).

T[n_,k_]:=Numerator[n(DivisorSigma[0,n]-1)/(DivisorSigma[1,n]-n/Part[Divisors[n],k])]; Table[T[n,k],{n,2,23},{k,DivisorSigma[0,n]}]//Flatten

T(n, k) = numerator(n*(tau(n) - 1)/(sigma(n) - n/A027750(n, k))).

cp's OEIS Frontend

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

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

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A250094 Positive integers k such that the numerator of the harmonic mean of the divisors of k is equal to k.

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A361316 Numerators of the harmonic means of the infinitary divisors of the positive integers.

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A361782 Numerators of the harmonic means of the bi-unitary divisors of the positive integers.

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A335369 Harmonic numbers k such that k*p is not a harmonic number for all the primes p that do not divide k.

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A348658 Numbers whose numerator and denominator of the harmonic mean of their divisors are both Fibonacci numbers.

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A348659 Numbers whose numerator and denominator of the harmonic mean of their divisors are both prime numbers.

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