A059248 -id:A059248 - cp's OEIS frontend

A247240 Numbers such that A059248(k), the numerator of Sum_{i=1..k} 1/Fibonacci(i), is not equal to A250744(k), the denominator of the harmonic mean of the first k positive Fibonacci numbers.

2, 7, 35, 245, 485, 12914

Offset: 1

Michel Marcus, Nov 28 2014

Next term > 20000. - Jinyuan Wang, Feb 14 2020

s=0; lst={}; Do[s+=1/Fibonacci[n]; If[Numerator[s]!=Denominator[n/s], AppendTo[lst, n]], {n, 10000}]; lst (* Jinyuan Wang, Feb 14 2020 *)

a(6) from Jinyuan Wang, Feb 14 2020

A035105 a(n) = LCM of Fibonacci sequence {F_1,...,F_n}.

Original entry on oeis.org

1, 1, 2, 6, 30, 120, 1560, 10920, 185640, 2042040, 181741560, 1090449360, 254074700880, 7368166325520, 449458145856720, 21124532855265840, 33735878969859546480, 640981700427331383120, 2679944489486672512824720, 109877724068953573025813520

Offset: 1

Fred Schwab (fschwab(AT)nrao.edu)

Alois P. Heinz, Table of n, a(n) for n = 1..124
Yuri V. Matiyasevich and Richard K. Guy, A new formula for Pi, The American Mathematical Monthly, Vol 93, No. 8 (1986), pp. 631-635.
Carlo Sanna, On the l.c.m. of shifted Fibonacci numbers, arXiv:2007.13330 [math.NT], 2020.
Index entries for sequences related to LCM's

Cf. A000045, A001622, A003266, A059248.

a:= proc(n) option remember; `if`(n=1, 1,
      ilcm(a(n-1), combinat[fibonacci](n)))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Feb 12 2018

a[ n_ ] := LCM@@Table[ Fibonacci[ k ], {k, 1, n} ]
With[{fibs=Fibonacci[Range[20]]},Table[LCM@@Take[fibs,n],{n, Length[ fibs]}]] (* Harvey P. Dale, Apr 29 2019 *)

a(n)=lcm(apply(fibonacci,[1..n])) \\ Charles R Greathouse IV, Oct 07 2016

from math import lcm
from sympy import fibonacci
def A035105(n): return lcm(*(fibonacci(i) for i in range(1,n+1))) # Chai Wah Wu, Jul 17 2022

log(a(n)) ~ 3*n^2*log(phi)/Pi^2, where phi is the golden ratio, or equivalently lim_{n->oo} sqrt(6*log(A003266(n))/log(a(n))) = Pi. - Amiram Eldar, Jan 30 2019

A059840 a(n) = F(n)F(n-1) if n odd otherwise F(n)F(n-1)-1, where F = Fibonacci numbers A000045.

Original entry on oeis.org

0, 0, 2, 5, 15, 39, 104, 272, 714, 1869, 4895, 12815, 33552, 87840, 229970, 602069, 1576239, 4126647, 10803704, 28284464, 74049690, 193864605, 507544127, 1328767775, 3478759200, 9107509824, 23843770274, 62423800997, 163427632719, 427859097159, 1120149658760, 2932589879120

Offset: 1

N. J. A. Sloane, Feb 26 2001

Harry J. Smith, Table of n, a(n) for n = 1..500
S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
H. Ohtsuka and S. Nakamura, On the sum of reciprocal sums of Fibonacci numbers, Fibonacci Quart. 46/47 (2008/2009), 153-159.
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000045, A001654, A059248, A064831, A119996.

List([1..30],n->Sum([1..n-2],k->Fibonacci(k)*Fibonacci(k+2))); # Muniru A Asiru, Aug 09 2018

F:=Fibonacci; [(n mod 2) eq 0 select F(n)*F(n-1)-1 else F(n)*F(n-1): n in [1..30]]; // G. C. Greubel, Jul 23 2019

seq(coeff(series(x^3*(2-x)/((1-x^2)*(1-3*x+x^2)), x,n+1),x,n),n=1..30); # Muniru A Asiru, Aug 09 2018

Table[If[OddQ[n],Fibonacci[n]Fibonacci[n-1],Fibonacci[n] Fibonacci[n-1]-1],{n,30}]  (* Harvey P. Dale, Apr 20 2011 *)

a(n) = { fibonacci(n)*fibonacci(n-1) - (n%2 == 0) } \\ Harry J. Smith, Jun 29 2009

a=(x^3*(2-x)/((1-x^2)*(1-3*x+x^2))).series(x, 30).coefficients(x, sparse=False); a[1:] # G. C. Greubel, Jul 23 2019

G.f.: x^3*(2 - x)/((1 - x^2)*(1 - 3*x + x^2)). See a comment on A080144. - Wolfdieter Lang, Jul 30 2012
a(n) = Sum_{k=1..n-2} F(k)*F(k+2). - Alexander Adamchuk, May 17 2007
a(n+2) = (3*A001654(n) + A027941(n))/2, n >= 0. - Wolfdieter Lang, Jul 21 2012
a(n+2) = (3*(-1)^(n+1) - 5 + 2*Lucas(2*n + 3))/10, n >= 0. - Ehren Metcalfe, Aug 21 2017
a(n) = floor(1/(Sum_{k>=n} 1/Fibonacci(k)^2)) [Ohtsuka and Nakamura]. - Michel Marcus, Aug 09 2018
For n > 2, 2 * A000217(a(n)) = A228873(n-2). - Diego Rattaggi, Jan 27 2021

A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).

Original entry on oeis.org

1, 5, 14, 39, 103, 272, 713, 1869, 4894, 12815, 33551, 87840, 229969, 602069, 1576238, 4126647, 10803703, 28284464, 74049689, 193864605, 507544126, 1328767775, 3478759199, 9107509824, 23843770273, 62423800997, 163427632718

Offset: 1

Alexander Adamchuk, Aug 03 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-3,1).

Cf. A000045, A059248, A064831, A001654, A045468, A064739, A091729.

F:=Fibonacci;; List([1..30], n-> F(n+1)*F(n+2)-1); # G. C. Greubel, Jul 23 2019

[Fibonacci(n+1)* Fibonacci(n+2)-1: n in [1..30]]; // Vincenzo Librandi, Aug 14 2012

with(combinat): seq(fibonacci(n+1)*fibonacci(n+2)-1, n=1..30); # Zerinvary Lajos, Jan 31 2008

Numerator[Table[Sum[1/(Fibonacci[k]*Fibonacci[k+2]),{k,n}],{n,30}]]
LinearRecurrence[{3,0,-3,1},{1,5,14,39},30] (* Harvey P. Dale, Aug 22 2011 *)

vector(30, n, f=fibonacci; f(n+1)*f(n+2)-1) \\ G. C. Greubel, Jul 23 2019

f=fibonacci; [f(n+1)*f(n+2)-1 for n in (1..30)] # G. C. Greubel, Jul 23 2019

a(n) = 3*a(n-1) - 3*a(n-3) + a(n-4); a(0)=1, a(1)=5, a(2)=14, a(3)=39. - Harvey P. Dale, Aug 22 2011
G.f.: ((x-2)*x-1)/(x^4 - 3*x^3 + 3*x - 1). - Harvey P. Dale, Aug 22 2011
a(n) = Fibonacci(n+1)*Fibonacci(n+2) - 1. - Gary Detlefs, Mar 31 2012
a(n) = Sum_{k=1..n} Fibonacci(k+1)^2. Can be proved by induction from Gary Detlefs formula. - Joel Courtheyn, Mar 15 2021

A059247 Denominator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).

Original entry on oeis.org

1, 2, 3, 1, 1, 3, 7, 2, 9, 10, 11, 12, 13, 14, 1, 8, 17, 9, 19, 10, 3, 11, 23, 2, 25, 2, 27, 28, 29, 10, 31, 32, 11, 34, 35, 9, 37, 19, 13, 20, 41, 1, 43, 1, 45, 23, 1, 8, 49, 50, 51, 52, 53, 54, 5, 56, 19, 58, 59, 20, 61, 62, 3, 8, 65, 33, 67, 17, 69, 35, 71

Offset: 1

N. J. A. Sloane, Jan 21 2001

			1, 3/2, 5/3, 2, 2, 7/3, 16/7, 5/2, 23/9, 27/10, ...

M. Aigner and G. M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 135.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A006218, A059246 and also A059248/A035105.

Denominator[Table[Sum[DivisorSigma[0, j]/n, {j,1,n}], {n,1,100}]] (* G. C. Greubel, Jan 02 2016 *)

a(n) = denominator(sum(j=1, n, numdiv(j))/n); \\ Michel Marcus, Jan 03 2017

from math import isqrt, gcd
def A059247(n): return n//gcd(n,(lambda m: 2*sum(n//k for k in range(1, m+1))-m*m)(isqrt(n))) # Chai Wah Wu, Oct 08 2021

a(n) = denominator(A006218(n)/n). - Michel Marcus, Jan 03 2017

A250744 Denominator of the harmonic mean of the first n positive Fibonacci numbers.

Original entry on oeis.org

1, 1, 5, 17, 91, 379, 721, 35849, 614893, 6800951, 607326679, 3651532639, 851897554247, 24724573280923, 301787157353771, 14188276949397301, 22662903194758542865, 430644772287132696121, 1800653989272587268758525, 369150309888695460837999593

Offset: 1

Colin Barker, Nov 27 2014

Similar to A059248. - Michel Marcus and Colin Barker, Nov 28 2014

			a(4) = 17 because the first 4 positive Fibonacci numbers are [1,1,2,3], and 4/(1/1+1/1+1/2+1/3) = 24/17.

Colin Barker, Table of n, a(n) for n = 1..120

Cf. A000045 (Fibonacci numbers), A250743 (numerators).

Cf. A059248.

Module[{nn=20,f},f=Fibonacci[Range[nn]];Table[HarmonicMean[Take[f,n]],{n,nn}]]//Denominator (* Harvey P. Dale, Aug 31 2020 *)

s=vector(30); f=Vec(x/(1-x-x^2)+O(x^(#s+1))); n=d=0; for(k=1, #s, n++; d+=1/f[k]; s[k]=denominator(n/d)); s

A225904 Numerator of Sum_{k=1..n} 1/L(k) where L(n) is the n-th Lucas number (A000204).

Original entry on oeis.org

1, 4, 19, 145, 1679, 5191, 153311, 7286005, 69689327, 2869226821, 572447760301, 6593608277800, 3438637721790797, 966842075996112436, 119933240206586434591, 264753799412041684949165, 945570749875765527295137611, 303554979754466691916744807193

Offset: 1

Michel Lagneau, May 20 2013

			1, 4/3, 19/12, 145/84, 1679/924, 5191/2772, ... = A225904/A225905.

Cf. A000204, A059248, A035105, A225905.

Numerator[Table[Sum[1/LucasL[k], {k, 1, n}], {n, 1, 30}]]
Numerator[Accumulate[1/LucasL[Range[20]]]] (* Harvey P. Dale, Sep 16 2016 *)

A225905 Denominator of Sum_{k=1..n} 1/L(k) where L(n) is the n-th Lucas number (A000204).

Original entry on oeis.org

1, 3, 12, 84, 924, 2772, 80388, 3778236, 35893242, 1471622922, 292852961478, 3367809056997, 1754628518695437, 493050613753417797, 61138276105423806828, 134932175364670341669396, 481842798227237790101413116, 154671538230943330622553610236

Offset: 1

Michel Lagneau, May 20 2013

			1, 4/3, 19/12, 145/84, 1679/924, 5191/2772, ... = A225904/A225905.

Cf. A000204, A059248, A035105, A225904.

Denominator[Table[Sum[1/LucasL[k], {k, 1, n}], {n, 1, 30}]]
Denominator[Accumulate[1/LucasL[Range[20]]]] (* Harvey P. Dale, Mar 11 2015 *)

cp's OEIS Frontend

A247240 Numbers such that A059248(k), the numerator of Sum_{i=1..k} 1/Fibonacci(i), is not equal to A250744(k), the denominator of the harmonic mean of the first k positive Fibonacci numbers.

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A035105 a(n) = LCM of Fibonacci sequence {F_1,...,F_n}.

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A059840 a(n) = F(n)*F(n-1) if n odd otherwise F(n)*F(n-1)-1, where F = Fibonacci numbers A000045.

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A119996 Numerator of Sum_{k=1..n} 1/(Fibonacci(k)*Fibonacci(k+2)).

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A059247 Denominator of Sum_{j=1..n} d(j)/n, where d = number of divisors function (A000005).

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A250744 Denominator of the harmonic mean of the first n positive Fibonacci numbers.

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A225904 Numerator of Sum_{k=1..n} 1/L(k) where L(n) is the n-th Lucas number (A000204).

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A059840 a(n) = F(n)F(n-1) if n odd otherwise F(n)F(n-1)-1, where F = Fibonacci numbers A000045.