A102928 -id:A102928 - cp's OEIS frontend

A368373 a(n) = denominator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

1, 6, 11, 50, 137, 98, 363, 1522, 7129, 14762, 83711, 172042, 1145993, 2343466, 1195757, 4873118, 42142223, 28548602, 275295799, 22334054, 18858053, 38186394, 444316699, 2695645910, 34052522467, 68791484534, 312536252003, 630809177806, 9227046511387, 18609365660294, 290774257297357

Offset: 1

N. J. A. Sloane, Jan 24 2024

			0, 1/6, 4/11, 29/50, 111/137, 103/98, 472/363, 2369/1522, 12965/7129, 30791/14762, 197346/83711, 452993/172042, 3337271/1145993, 7485915/2343466, 4160656/1195757, 18358463/4873118, ...

Paolo Xausa, Table of n, a(n) for n = 1..2000

Cf. A102928/A175441, A368366, A368372.

AM:=proc(n) local i; (add(i,i=1..n)/n); end;
HM:=proc(n) local i; (add(1/i,i=1..n)/n)^(-1); end;
s1:=[seq(AM(n)-HM(n),n=1..50)];

A368373[n_] := Denominator[(n+1)/2 - n/HarmonicNumber[n]];
Array[A368373, 35] (* Paolo Xausa, Jan 29 2024 *)

a368373(n) = denominator((n+1)/2 - n/harmonic(n)) \\ Hugo Pfoertner, Jan 25 2024

from fractions import Fraction
from itertools import count, islice
def agen(): # generator of terms
    A = H = 0
    for n in count(1):
        A += n
        H += Fraction(1, n)
        yield ((A*Fraction(1, n) - n/H)).denominator
print(list(islice(agen(), 31))) # Michael S. Branicky, Jan 24 2024

from fractions import Fraction
from sympy import harmonic
def A368373(n): return (Fraction(n+1,2)-Fraction(n,harmonic(n))).denominator # Chai Wah Wu, Jan 25 2024

A187487 Numerator of n minus n-th harmonic number.

Original entry on oeis.org

0, 1, 7, 23, 163, 71, 617, 1479, 15551, 17819, 221209, 246619, 3538687, 3873307, 4209643, 9094961, 166145857, 59239139, 1199057081, 254554945, 89778475, 94716499, 2292289173, 7218943253, 189040347533

Offset: 1

Alonso del Arte, Mar 10 2011

Robert Israel, Table of n, a(n) for n = 1..2286

Denominators are in A002805.

Cf. A001008 (numerators of the harmonic numbers). Adding this sequence to that sequence gives A102928.

f:=n -> numer(n - harmonic(n)):map(f, [$1..100]); # Robert Israel, Apr 26 2021

Numerator[Table[n - HarmonicNumber[n], {n, 25}]]

a(n) = numerator(n - sum(i=1, n, 1/i)); \\ Michel Marcus, Apr 26 2021

a(n) = n - Sum_{i = 1..n} 1/i.

a(n) = Sum_{i = 2..n} (i - 1)/i.

A132773 a(n) = n*(n + 31).

Original entry on oeis.org

0, 32, 66, 102, 140, 180, 222, 266, 312, 360, 410, 462, 516, 572, 630, 690, 752, 816, 882, 950, 1020, 1092, 1166, 1242, 1320, 1400, 1482, 1566, 1652, 1740, 1830, 1922, 2016, 2112, 2210, 2310, 2412, 2516, 2622, 2730, 2840, 2952, 3066, 3182, 3300, 3420, 3542, 3666

Offset: 0

Omar E. Pol, Aug 28 2007

Felix P. Muga II, Extending the Golden Ratio and the Binet-de Moivre Formula, Preprint on ResearchGate, March 2014.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001008, A002378, A005563, A028347, A028552, A028557, A028560, A028563, A028566, A028569.
Cf. A098603, A098847, A098848, A098849, A098850, A102928, A120071, A132758, A132759, A132760.
Cf. A132761, A132762, A132763, A132764, A132765, A132766, A132767, A132768, A132769, A132770.
Cf. A132771, A132772.

Table[n*(n + 31), {n, 0, 100}] (* Wesley Ivan Hurt, Apr 16 2024 *)

a(n)=n*(n+31) \\ Charles R Greathouse IV, Jun 17 2017

G.f.: 2*x*(-16+15*x)/(-1+x)^3. - R. J. Mathar, Nov 14 2007
a(n) = 2*A132758(n). - R. J. Mathar, Jul 22 2009
a(n) = 2*n + a(n-1) + 30, with n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
From Amiram Eldar, Jan 16 2021: (Start)
Sum_{n>=1} 1/a(n) = H(31)/31 = A001008(31)/A102928(31) = 290774257297357/2238255069850800, where H(k) is the k-th harmonic number.
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*log(2)/31 - 7313175618421/319750724264400. (End)
From Elmo R. Oliveira, Dec 13 2024: (Start)
E.g.f.: exp(x)*x*(32 + x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

A216101 Primes which are the integer harmonic mean of the previous prime and the following prime.

Original entry on oeis.org

13, 19, 43, 47, 83, 89, 103, 109, 131, 167, 193, 229, 233, 313, 349, 353, 359, 383, 389, 409, 443, 449, 463, 503, 643, 647, 677, 683, 691, 709, 797, 823, 859, 883, 919, 941, 971, 983, 1013, 1093, 1097, 1109, 1171, 1193, 1217, 1279, 1283, 1303, 1373, 1429, 1433

Offset: 1

César Eliud Lozada, Sep 01 2012

nonn

The harmonic mean of N numbers p1,p2,..,pN is the number H whose reciprocal is the arithmetic mean of the reciprocals of p1,p2,..,pN; that is to say, 1/H = ((1/p1)+(1/p2)+..+(1/pN))/N.
So, for two quantities p1 and p2, their harmonic mean may be written as H=(2*p1*p2)/(p1+p2).
The harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

			The primes before and after the prime p=13 are p1=11 and p2=17. So, the harmonic mean of p1 and p2 is 2*11*17/(11+17)=13.35714285... whose integer part is p=13. Then p=13 belongs to the sequence.
The primes before and after the prime p=17 are p1=13 and p2=19. The harmonic mean of p1 and p2 is 2*13*19/(13+19)=15.4375, having 15 as its integer part. Therefore, as 15<>p, p=17 is not in the sequence.

Wikipedia, Harmonic Mean

Cf. A006562, A001008, A102928, A216098, A216124.

A := {}: for n from 2 to 1000 do p1 := ithprime(n-1): p := ithprime(n): p2 := ithprime(n+1): if p = floor(2*p1*p2/(p1+p2)) then A := `union`(A, {p}) end if end do; A := A;

t = {}; Do[p = Prime[n]; If[Floor[HarmonicMean[{Prime[n - 1], Prime[n + 1]}]] == p, AppendTo[t, p]], {n, 2, 200}]; t (* T. D. Noe, Sep 04 2012 *)

A356137 Positive integers m such that the fractional part of the geometric mean of the sequence s(m) does not exceed the fractional part of the arithmetic mean of s(m), where s(m) is the sequence 1 + 1/1, 2 + 1/2, ..., m + 1/m.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 10, 13, 14, 16, 18, 22, 24, 26, 30, 32, 34, 38, 40, 42, 46, 48, 54, 56, 61, 62, 64, 69, 70, 72, 78, 80, 86, 88, 92, 94, 96, 100, 102, 108, 110, 115, 116, 118, 124, 126, 132, 134, 138, 140, 146, 148, 154, 156, 161, 162, 164, 170, 172, 178, 180

Offset: 1

Mike Jones, Jul 27 2022

nonn

The idea is to take note of when the fractional parts of the geometric mean and arithmetic mean "follow suit" with respect to the celebrated geometric mean <= arithmetic mean inequality.

			2 is a term because the geometric mean of 1 + 1/1 and 2 + 1/2 is the geometric mean of 2 and 2.5, which is a bit less than 2.24, whereas the arithmetic mean of 2 and 2.5 is 2.25, and 0.24 <= 0.25.
4 is not a term because the geometric mean is 2.90..., whereas the arithmetic mean is 3.02..., and 0.90 > 0.02.

Wikipedia, Fractional part
Wikipedia, Inequality of arithmetic and geometric means

Cf. A356142/A102928 (the arithmetic mean of s(n)).

max=180; a={}; s[m_]:=m+1/m; For[m=1,m<=max,m++,If[FractionalPart[Mean[Table[s[k],{k,m}]]] >= FractionalPart[GeometricMean[Table[s[k],{k,m}]]],AppendTo[a,m]]]; a (* Stefano Spezia, Jul 27 2022 *)

isok(m) = my(v=vector(m, k, k+1/k)); frac(sqrtn(vecprod(v), m)) <= frac(vecsum(v)/m); \\ Michel Marcus, Jul 28 2022

More terms from Stefano Spezia, Jul 27 2022

A356142 a(n) is the reduced numerator of the arithmetic mean of the first n terms of the sequence [n + 1/n, n > 0].

Original entry on oeis.org

2, 9, 47, 145, 1037, 469, 4283, 10841, 120529, 145981, 1913231, 2248181, 33938753, 39009533, 44438957, 100454479, 1916734943, 712651981, 15018824599, 662986195, 1213859861, 1327904701, 33283587163, 108432400555, 2934259832467, 3166619637067, 30671014001603, 32922658468103

Offset: 1

Stefano Spezia, Jul 27 2022

Cf. A102928 (denominator), A356137.

Table[Numerator[Mean[Table[k+1/k,{k,n}]]],{n,28}]

a(n) = numerator(sum(k=1, n, k+1/k)/n); \\ Michel Marcus, Jul 28 2022

cp's OEIS Frontend

A368373 a(n) = denominator of AM(n)-HM(n), where AM(n) and HM(n) are the arithmetic and harmonic means of the first n positive integers.

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A187487 Numerator of n minus n-th harmonic number.

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A132773 a(n) = n*(n + 31).

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A216101 Primes which are the integer harmonic mean of the previous prime and the following prime.

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A356137 Positive integers m such that the fractional part of the geometric mean of the sequence s(m) does not exceed the fractional part of the arithmetic mean of s(m), where s(m) is the sequence 1 + 1/1, 2 + 1/2, ..., m + 1/m.

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A356142 a(n) is the reduced numerator of the arithmetic mean of the first n terms of the sequence [n + 1/n, n > 0].

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