A079586 -id:A079586 - cp's OEIS frontend

A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 2, 3, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 4, 1, 3, 2, 2, 2, 3, 1, 2, 3, 4, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 3, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 3, 3, 3, 3, 1, 3, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 4, 2, 2, 1, 4, 2, 2, 2, 3, 1, 4, 2, 2, 2, 2, 2, 4, 1, 2, 2, 3, 1, 4, 1, 4, 4

Offset: 1

Antti Karttunen, Oct 09 2017

nonn

			For n = 55, its proper divisors are [1, 5, 11], of which only two, namely 1 and 5 are in A000045, thus a(55) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10946

Cf. A000045, A005086, A010056, A079586, A293436.

Cf. also A293433.

With[{s = Fibonacci@ Range[2, 40]}, Table[DivisorSum[n, 1 &, And[MemberQ[s, #], # != n] &], {n, 105}]] (* Michael De Vlieger, Oct 09 2017 *)

A010056(n) = { my(k=n^2); k+=(k+1)<<2; (issquare(k) || (n>0 && issquare(k-8))) }; \\ This function from Charles R Greathouse IV, Jul 30 2012
A293435(n) = sumdiv(n,d,(dA010056(d));

a(n) = Sum_{d|n, dA010056(d).
a(n) = A005086(n) - A010056(n).
G.f.: Sum_{k>=2} x^(2*Fibonacci(k)) / (1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Apr 14 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - Amiram Eldar, Jul 05 2025

A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.

Original entry on oeis.org

1, 3, 5, 7, 9, 12, 13, 16, 18, 21, 22, 25, 27, 29, 32, 35, 36, 39, 40, 43, 46, 48, 49, 53, 55, 58, 60, 62, 63, 67, 68, 71, 73, 76, 78, 81, 82, 84, 87, 91, 92, 96, 97, 99, 102, 104, 105, 109, 110, 113, 115, 118, 119, 122, 125, 128, 130, 132, 133, 137, 138, 140, 143, 146

Offset: 1

Antreas P. Hatzipolakis (xpolakis(AT)hol.gr)

			a(15)=[ 15/1 ]+[ 15/2 ]+[ 15/3 ]+[ 15/5 ]+[ 15/8 ]+[ 15/13 ]+[ 15/21 ]+...=32.

Vaclav Kotesovec, Plot of a(n)/n for n = 1..100000

Cf. A005086.

[&+[Floor(n/Fibonacci(k+2)):k in [0..n]]:n in [1..64]]; // Marius A. Burtea, Jul 16 2019

with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k)), k=2..15)) od:

Table[Sum[Floor[n/Fibonacci[k] ],{k,2,200}],{n,70}] (* Harvey P. Dale, Jul 21 2021 *)
Table[Sum[Floor[n/Fibonacci[k]], {k, 2, Log[Sqrt[5]*n]/Log[GoldenRatio] + 1}], {n, 1, 100}] (* Vaclav Kotesovec, Aug 30 2021 *)

G.f.: (1/(1 - x)) * Sum_{k>=2} x^Fibonacci(k)/(1 - x^Fibonacci(k)). - Ilya Gutkovskiy, Jul 16 2019

Conjecture: a(n) ~ c * n, where c = A079586 - 1. - Vaclav Kotesovec, Aug 30 2021

More terms from Simon Plouffe, who points out that the first differences give A005086

More terms from James Sellers, Feb 19 2001

A105393 Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.

Original entry on oeis.org

2, 4, 2, 6, 3, 2, 0, 7, 5, 1, 1, 6, 7, 2, 4, 1, 1, 8, 7, 7, 4, 1, 5, 6, 9, 4, 1, 2, 9, 2, 6, 6, 2, 0, 3, 7, 4, 3, 2, 0, 2, 5, 9, 7, 7, 4, 5, 1, 3, 8, 3, 0, 9, 0, 5, 1, 1, 0, 1, 0, 2, 8, 3, 4, 5, 4, 6, 6, 1, 1, 9, 3, 7, 5, 1, 1, 1, 9, 7, 8, 6, 3, 6, 8, 7, 7, 5, 3, 8, 9, 8, 1, 5, 2, 1, 5, 3, 6, 3, 6, 3, 7, 9, 2, 1

Offset: 1

Jonathan Vos Post, Apr 04 2005

Known to be transcendental. - Benoit Cloitre, Jan 07 2006
Compare with Sum_{n >= 1} 1/(F(n)^2 + 1) = (5*sqrt(5) - 3)/6 and Sum_{n >= 3} 1/(F(n)^2 - 1) = (43 - 15*sqrt(5))/18. - Peter Bala, Nov 19 2019
Duverney et al. (1997) proved that this constant is transcendental. - Amiram Eldar, Oct 30 2020

			2.426320751167241187741569...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Michel Waldschmidt, Elliptic functions and transcendence, in: Krishnaswami Alladi (ed.), Surveys in number theory, Springer, New York, NY, 2008, pp. 143-188, alternative link. See Corollary 51.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Index entries for transcendental numbers

Cf. A000045, A007598 (squares of Fibonacci numbers).

Cf. A079586, A093540, A105394.

RealDigits[Total[1/Fibonacci[Range[500]]^2],10,120][[1]] (* Harvey P. Dale, May 31 2016 *)

sum(k=1,500,1./fibonacci(k)^2) \\ Benoit Cloitre, Jan 07 2006

Equals Sum_{k>=1} 1/F(k)^2 = 2.4263207511672411877... - Benoit Cloitre, Jan 07 2006

More terms from Benoit Cloitre, Jan 07 2006

A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).

Original entry on oeis.org

2, 8, 9, 1, 4, 4, 6, 4, 8, 5, 7, 0, 6, 7, 1, 5, 8, 3, 1, 1, 2, 3, 0, 5, 5, 0, 9, 6, 1, 5, 7, 2, 9, 1, 6, 6, 9, 5, 4, 8, 8, 1, 9, 5, 1, 5, 8, 9, 6, 9, 1, 3, 6, 0, 0, 2, 5, 0, 2, 6, 4, 8, 5, 0, 6, 3, 0, 3, 5, 7, 6, 1, 7, 3, 8, 8, 6, 4, 5, 5, 1, 5, 8, 2, 4, 1, 1, 5, 8, 3, 1, 8, 2, 8, 5

Offset: 0

Michel Lagneau, Mar 26 2011

André-Jeannin (1989) proved that this constant is irrational, and Tachiya (2004) proved that it does not belong to the quadratic number field Q(sqrt(5)). - Amiram Eldar, Oct 30 2020

			0.2891446485706715831123055096157291669...

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, Vol. 308, No. 19 (1989), pp. 539-541.
Yohei Tachiya, Irrationality of certain Lambert series, Tokyo Journal of Mathematics, Vol. 27, No. 1 (2004), pp. 75-85.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.

Cf. A000045, A001622, A079586.

Cf. A153386, A153387, A324007.

with(combinat, fibonacci):Digits:=100:s:=0:for n from 1 to 2000 do: a1:=fibonacci(n):s:=s+evalf(1/a1)*(-1)^(n+1):od:print(s):

digits = 95; NSum[(-1)^(n+1)*(1/Fibonacci[n]), {n, 1, Infinity}, WorkingPrecision -> digits+1, NSumTerms -> digits] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Jan 28 2014 *)

-sumalt(n=1,(-1)^n/fibonacci(n)) \\ Charles R Greathouse IV, Oct 03 2016

Equals sqrt(5) * Sum_{k>=0} (-1)^k/(phi^(2*k+1) + (-1)^k), where phi is the golden ratio (A001622). - Amiram Eldar, Oct 04 2020
Equals A153387 - A153386. - Joerg Arndt, Oct 04 2020
Equals 1 - A324007. - Amiram Eldar, Feb 09 2023

Offset corrected by Arkadiusz Wesolowski, Jun 28 2011

A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.

Original entry on oeis.org

1, 2, 0, 7, 2, 9, 1, 9, 9, 6, 9, 8, 5, 7, 4, 7, 0, 7, 4, 4, 1, 7, 2, 0, 4, 1, 8, 4, 2, 5, 7, 6, 9, 9, 9, 4, 5, 3, 0, 6, 9, 2, 1, 4, 5, 4, 0, 1, 9, 0, 3, 6, 3, 7, 6, 9, 5, 1, 3, 1, 1, 5, 9, 4, 2, 2, 1, 2, 2, 4, 0, 0, 1, 5, 4, 0, 7, 0, 3, 5, 7, 7, 6, 1, 6, 7, 7, 6, 5, 5, 9, 7, 8, 6, 8, 8, 9, 9, 9, 2

Offset: 1

Jonathan Vos Post, Apr 04 2005

This constant is transcendental (Duverney et al., 1997). - Amiram Eldar, Oct 30 2020

			1.207291996985747074417204...

Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Wiley, 1987, p. 97.

Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, C. R. Acad. Sci. Paris Ser. I Math., Vol. 308, No. 19 (1989), pp. 539-541.
Paul S. Bruckman,, Problem H-347, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. (1982), p. 372; It All Adds Up, Solution to Problem H-347 by the proposer, ibid., Vol. 22, No. 1 (1984), pp. 94-96.
Daniel Duverney, Keiji Nishioka, Kumiko Nishioka and Iekata Shiokawa, Transcendence of Rogers-Ramanujan continued fraction and reciprocal sums of Fibonacci numbers, Proceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 73, No. 7 (1997), pp. 140-142.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Fibonacci Number.
Eric Weisstein's World of Mathematics, Reciprocal Fibonacci Constant.
Index entries for transcendental numbers

Cf. A000032, A001254 (squares of Lucas numbers).

Cf. A079586, A093540, A105393, A153415.

f[n_] := f[n] = RealDigits[ Sum[ 1/LucasL[k]^2, {k, 1, n}], 10, 100] // First; f[n=100]; While[f[n] != f[n-100], n = n+100]; f[n] (* Jean-François Alcover, Feb 13 2013 *)

Equals Sum_{n >= 1} 1/L(n)^2.
Equals (1/8)*( theta_3(beta)^4 - 1 ), where beta = (3 - sqrt(5))/2 and theta_3(q) = 1 + 2*Sum_{n >= 1} q^(n^2) is a theta function. See Borwein and Borwein, Exercise 7(f), p. 97. - Peter Bala, Nov 13 2019
Equals c*(2*c+1), where c = A153415 (follows from the identity Sum_{n=-oo..oo} 1/L(n^2) = (Sum_{n=-oo..oo} 1/L(2*n))^2, see Bruckman, 1982). - Amiram Eldar, Jan 27 2022

A228040 Decimal expansion of sum of reciprocals, row 2 of Wythoff array, W = A035513.

Original entry on oeis.org

6, 2, 9, 5, 2, 4, 8, 3, 9, 8, 7, 6, 3, 1, 2, 4, 4, 9, 5, 3, 5, 4, 6, 1, 7, 9, 5, 3, 4, 1, 8, 5, 0, 1, 9, 3, 3, 1, 6, 2, 5, 9, 6, 8, 3, 8, 2, 8, 8, 8, 6, 0, 8, 7, 7, 9, 7, 3, 8, 1, 9, 0, 7, 0, 8, 3, 7, 2, 8, 2, 7, 4, 2, 1, 3, 1, 2, 7, 0, 4, 4, 6, 4, 5, 7, 0

Offset: 0

Clark Kimberling, Aug 05 2013

Let c be the constant given by A079586, that is, the sum of reciprocals of the Fibonacci numbers F(k) for k>=1. The number c-1, the sum of reciprocals of row 1 of W, is known to be irrational (see A079586). Conjecture: the same is true for all the other rows of W.

Let h be the constant given at A153387 and s(n) the sum of reciprocals of numbers in row n of W. Then h < 1 + s(n)*floor(n*tau) < c. Thus, s(n) -> 0 as n -> oo.

			1/4 + 1/7 + 1/11 + ... = 0.629524839876312449535461795341...

Cf. A035513, A079586, A093540, A228041, A228042, A228043.

f[n_] := f[n] = Fibonacci[n]; g = GoldenRatio; w[n_, k_] := w[n, k] = f[k + 1]*Floor[n*g] + f[k]*(n - 1);
n = 2; Table[w[n, k], {n, 1, 5}, {k, 1, 5}]
r = N[Sum[1/w[n, k], {k, 1, 2000}], 120]
RealDigits[r, 10]

Equals A093540 - 4/3. - Amiram Eldar, May 22 2021

A350902 a(n) = (5F(n)F(n-1)F(2n-1)a(n-1) + F(n-1)L(n)a(n-2))/(L(n-1)F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

1, 0, 3, 25, 816, 59475, 12031005, 6229446000, 8517168411895, 30387269735449725, 284188952072106783648, 6954889250543118311091775, 445684855849546942072130113089, 74767094861864103592878982016253600, 32838249084789127737424410920015676309123

Offset: 0

Amiram Eldar, Jan 21 2022

Although the recurrence relation involves fractions, all the terms are integers.
The sequence of fractions b(n) = A350903(n)/A350904(n) is defined by the same recurrence relation, but with the initial terms 0 and 1 instead of 1 and 0.
André-Jeannin (1991) used this sequence and the sequence b(n) to prove that s = Sum_{n>=1} 1/F(n) (A079586) is an irrational number.
The sequence of ratios r(n) = b(n)/a(n) rapidly converges to s. For example, abs(r(16)-s) < 10^(-100) and abs(r(49)-s) < 10^(-1000).

Amiram Eldar, Table of n, a(n) for n = 0..69
Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350903, A350904.

With[{F = Fibonacci, L = LucasL}, a[0] = 1; a[1] = 0; a[n_] := a[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*a[n - 1] + F[n - 1]*L[n]*a[n - 2])/(L[n - 1]*F[n]); Array[a, 15, 0]]

Limit_{n->oo} A350903(n)/(A350904(n)*a(n)) = A079586 (André-Jeannin, 1991).

A350903 Numerators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

0, 1, 10, 84, 8225, 999146, 161691205, 4081394133187, 801267937794945, 451272063930179690869, 955797228958312695758495, 12869303093903467063139191673469, 141131682569461636438244407470674215, 5214528077594695050414454970728001934806021

Offset: 0

Amiram Eldar, Jan 21 2022

See A350902 for details.

			The sequence of fractions begins with 0, 1, 10, 84, 8225/3, 999146/5, 161691205/4, 4081394133187/195, 801267937794945/28, 451272063930179690869/4420, ...

Amiram Eldar, Table of n, a(n) for n = 0..60
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350902, A350904 (denominators).

With[{F = Fibonacci, L = LucasL}, u[0] = 0; u[1] = 1; u[n_] := u[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*u[n - 1] + F[n - 1]*L[n]*u[n - 2])/(L[n - 1]*F[n]); Numerator @ Array[u, 15, 0]]

A350904 Denominators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

Original entry on oeis.org

1, 1, 1, 1, 3, 5, 4, 195, 28, 4420, 1001, 550732, 94248, 20757737, 150585864, 596098336680, 84878386593, 17090110926980520, 1216260982575912, 13296541287045886485, 484071647034823848, 3418959485072391296664264, 19630886922468003512297

Offset: 0

Amiram Eldar, Jan 21 2022

See A350902 for details.

Amiram Eldar, Table of n, a(n) for n = 0..60
Richard André-Jeannin, Sequences of Integers Satisfying Recurrence Relations, The Fibonacci Quarterly, Vol. 29, No. 3 (1991), pp. 205-208.

Cf. A000032, A000045, A079586, A350902, A350903 (numerators).

With[{F = Fibonacci, L = LucasL}, u[0] = 0; u[1] = 1; u[n_] := u[n] = (5*F[n]*F[n - 1]*F[2*n - 1]*u[n - 1] + F[n - 1]*L[n]*u[n - 2])/(L[n - 1]*F[n]); Denominator @ Array[u, 25, 0]]

A360957 Decimal expansion of Sum_{i>=1 and i!=0 (mod 3)} 1/Fibonacci(i).

Original entry on oeis.org

2, 6, 9, 6, 3, 8, 3, 5, 2, 7, 3, 1, 0, 1, 4, 9, 3, 5, 6, 0, 3, 6, 1, 3, 0, 2, 0, 6, 9, 6, 8, 9, 3, 3, 8, 8, 3, 9, 1, 3, 6, 3, 8, 8, 8, 2, 1, 0, 7, 4, 3, 8, 9, 5, 8, 1, 9, 2, 4, 4, 5, 3, 8, 9, 6, 4, 4, 8, 0, 1, 1, 5, 8, 5, 8, 2, 4, 2, 0, 0, 3, 3, 0, 9, 6, 0, 6, 1, 6, 7, 7, 1, 1, 3, 2, 4, 9, 2, 3, 6, 3, 4, 3, 8, 1

Offset: 1

Kevin Ryde, Feb 28 2023

Sum of reciprocals of all odd Fibonacci numbers, so Sum_{j>=0} 1/A014437(j)

			2.6963835273101493560361302069689338...

Kevin Ryde, Table of n, a(n) for n = 1..10000

Cf. A014437, A079586, A153387, A360958.

Equals A079586 - A360958.

cp's OEIS Frontend

A293435 a(n) is the number of the proper divisors of n that are Fibonacci numbers (A000045).

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A038663 [ n/F_2 ] + [ n/F_3 ] + [ n/F_4 ] +..., F_n=Fibonacci numbers.

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A105393 Decimal expansion of sum of reciprocals of squares of Fibonacci numbers.

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A158933 Decimal expansion of Sum_{n>=1} ((-1)^(n+1))/F(n) where F(n) is the n-th Fibonacci number A000045(n).

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A105394 Decimal expansion of sum of reciprocals of squares of Lucas numbers.

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A228040 Decimal expansion of sum of reciprocals, row 2 of Wythoff array, W = A035513.

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A350902 a(n) = (5*F(n)*F(n-1)*F(2*n-1)*a(n-1) + F(n-1)*L(n)*a(n-2))/(L(n-1)*F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

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A350902 a(n) = (5F(n)F(n-1)F(2n-1)a(n-1) + F(n-1)L(n)a(n-2))/(L(n-1)F(n)), with a(0) = 1, a(1) = 0, where F(n) = A000045(n) and L(n) = A000032(n).

A350903 Numerators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).

A350904 Denominators of the sequence of fractions defined by u(n) = ((5F(n)F(n-1)F(2n-1)u(n-1) + F(n-1)L(n)u(n-2))/(L(n-1)F(n))), with u(0) = 0 and u(1) = 1, where F(n) = A000045(n) and L(n) = A000032(n).