A130233 -id:A130233 - cp's OEIS frontend

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 53, 89, 131, 139, 199, 233, 521, 607, 953, 1453, 1597, 2207, 2351, 2579, 3571, 6763, 9349, 10891, 28513, 28649, 28657, 42187, 44771, 46279, 75017, 189653, 317777, 514229, 1981891, 2177699, 3010349, 3206767

Offset: 1

T. D. Noe, Oct 17 2005

nonn

The difference F(i)-F(j) equals the sum F(j-1)+...+F(i-2) [Corrected by Patrick Capelle, Mar 01 2008]. In general, we need gcd(i,j)=1 for F(i)-F(j) to be prime. The exceptions are handled by the following rule: if i and j are both even or both odd, then F(i)-F(j) is prime if either (1) i-j=4 and L(i-2) is a Lucas prime or (2) i-j=2 and F(i-1) is a Fibonacci prime.

			The prime 139 is here because it is F(12)-F(5).

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000045 (Fibonacci numbers), A001605 (Fibonacci(n) is prime), A001606 (Lucas(n) is prime), A113189 (number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3).

lst={}; Do[p=Fibonacci[n]-Fibonacci[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, n-1}]; Union[lst]
Select[Union[Flatten[Differences/@Subsets[Fibonacci[Range[50]],{2}]]],PrimeQ] (* Harvey P. Dale, Aug 04 2024 *)

list(lim)=my(v=List(),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); if(isprime(s), listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

Original entry on oeis.org

0, 0, 0, 2, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Inverse of the Lucas sequence (A000032), nearly, since a(Lucas(n))=n except for n=1 (see A130241 and A130247 for other versions). For n>=2, a(n+1) is equal to the partial sum of the Lucas indicator sequence (see A102460).

			a(10)=5, since Lucas(5)=11>=10 but Lucas(4)=7<10.

G. C. Greubel, Table of n, a(n) for n = 0..5000

For partial sums see A130244.
Other related sequences: A000032, A130241, A130245, A130247, A130250, A130256, A130260.
Indicator sequence A102460.
Fibonacci inverse see A130233 - A130240, A104162.

Join[{0, 0, 0}, Table[Ceiling[Log[GoldenRatio, n + 1/2]], {n, 2, 50}]] (* G. C. Greubel, Dec 24 2017 *)

from itertools import islice, count
def A130242_gen(): # generator of terms
    yield from (0,0,0,2)
    a, b = 3, 4
    for i in count(3):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130242_list = list(islice(A130242_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) = ceiling(log_phi((n+sqrt(n^2-4))/2))=ceiling(arccosh(n/2)/log(phi)) where phi=(1+sqrt(5))/2.
a(n) = A130241(n-1) + 1 = A130245(n-1) for n>=3.
G.f.: x/(1-x)*(2x^2+sum{k>=2, x^Lucas(k)}).
a(n) = ceiling(log_phi(n-1/2)) for n>=3, where phi is the golden ratio.

A130259 Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 0

Hieronymus Fischer, May 25 2007, Jul 02 2007

nonn

Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130260 for another version).

a(n)+1 is the number of even Fibonacci numbers (A001906) <=n.

			a(10)=3 because A001906(3)=8<=10, but A001906(4)=21>10.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. partial sums A130261. Other related sequences: A000045, A001519, A130233, A130237, A130239, A130255, A130260, A104160. Lucas inverse: A130241 - A130248.

[Floor(Log((Sqrt(5)*n+1))/(2*Log((1+Sqrt(5))/2))): n in [0..100]]; // G. C. Greubel, Sep 12 2018

Table[Floor[1/2*Log[GoldenRatio, (Sqrt[5]*n + 1)]], {n, 0, 100}] (* G. C. Greubel, Sep 12 2018 *)

vector(100, n, n--; floor(log((sqrt(5)*n+1))/(2*log((1+sqrt(5))/2) ))) \\ G. C. Greubel, Sep 12 2018

a(n) = floor(arcsinh(sqrt(5)*n/2)/(2*log(phi))), where phi=(1+sqrt(5))/2.
a(n) = A130260(n+1) - 1.
G.f.: g(x) = 1/(1-x)*Sum_{k>=1} x^Fibonacci(2*k).
a(n) = floor(1/2*log_phi(sqrt(5)*n+1)) for n>=0.

A130261 Partial sums of the 'lower' even Fibonacci Inverse A130259.

Original entry on oeis.org

0, 1, 2, 4, 6, 8, 10, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 192, 197, 202, 207, 212, 217

Offset: 0

Hieronymus Fischer, May 25 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000045, A001519, A001906, A130233, A130235, A130236, A130255, A130257, A104162. Lucas inverse: A130241 - A130248.

[(&+[Floor(Log((Sqrt(5)*k+1))/(2*Log((1+Sqrt(5))/2))): k in [0..n]]): n in [0..50]]; // G. C. Greubel, Sep 12 2018

Table[Sum[Floor[1/2*Log[GoldenRatio, (Sqrt[5]*k + 1)]], {k, 0, n}], {n, 0, 50}] (* G. C. Greubel, Sep 12 2018 *)

for(n=0,50, print1(sum(k=0,n, floor(log((sqrt(5)*k+1))/(2*log((1 +sqrt(5))/2)))), ", ")) \\ G. C. Greubel, Sep 12 2018

a(n) = (n+1)*A130259(n) - A001519(A130259(n)+1) + 1.
a(n) = (n+1)*A130259(n) - Fib(2*A130259(n)+1) + 1.
G.f.: g(x) = 1/(1-x)^2*Sum_{k>=1} x^Fib(2*k).

A271354 Products of two distinct Fibonacci numbers, both greater than 1.

Original entry on oeis.org

6, 10, 15, 16, 24, 26, 39, 40, 42, 63, 65, 68, 102, 104, 105, 110, 165, 168, 170, 178, 267, 272, 273, 275, 288, 432, 440, 442, 445, 466, 699, 712, 714, 715, 720, 754, 1131, 1152, 1155, 1157, 1165, 1220, 1830, 1864, 1869, 1870, 1872, 1885, 1974, 2961, 3016

Offset: 1

Clark Kimberling, May 02 2016

For n > 5, the numbers F(i)*F(j) satisfying F(n-1) <= F(i)*F(j) <= F(n) also satisfy F(n-1) < F(i)*F(j) < F(n). They are the numbers for which i + j = n + 1, where 2 < i < j, so that the number of such F(i)*F(j) is floor(n/2) - 2. The least is 3*F(n-3) and the greatest is 2*F(n-2).

			2*3 = 6, 2*5 = 10, 3*5 = 15, 2*8 = 16.

Clark Kimberling, Table of n, a(n) for n = 1..1000
Clark Kimberling, Orderings of products of Fibonacci numbers, Fibonacci Quarterly 42:1 (2004), pp. 28-35.

Cf. A000045, A004526, A094565, A271356 (difference sequence), subsequence of A049997.

z = 200; f[n_] := Fibonacci[n];
Take[Sort[Flatten[Table[f[m] f[n], {n, 3, z}, {m, 3, n - 1}]]], 100]
Times@@@Subsets[Fibonacci[Range[3,20]],{2}]//Union (* Harvey P. Dale, Jul 12 2025 *)

list(lim)=my(v=List,F=vector(A130233(lim\2),k,fibonacci(k)),t); for(i=2,#F, for(j=1,i-1, t=F[i]*F[j]; if(t>lim,break); listput(v,t))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A004526(n) = number of numbers a(k) between F(n+3) and F(n+4), where F = A000045 (Fibonacci numbers).

A130239 Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

Original entry on oeis.org

0, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Hieronymus Fischer, May 17 2007, May 28 2007

nonn

			a(10) = 4 since Fib(4)^2 = 9 <= 10 but Fib(5)^2 = 25 > 10.

G. C. Greubel, Table of n, a(n) for n = 0..5000

Partial sums: A130240. Other related sequences: A000045, A130233, A130234, A130235, A130236, A130237, A130238, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

A130233:= func< n | Floor(Log(3/2 + n*Sqrt(5))/Log((1+Sqrt(5))/2)) >;
[A130233(Floor(Sqrt(n))): n in [0..120]]; // G. C. Greubel, Mar 18 2023

A130233[n_]:= Floor[Log[GoldenRatio, 3/2 +n*Sqrt[5]]];
Table[A130233[Floor[Sqrt[n]]], {n, 0, 120}] (* G. C. Greubel, Mar 18 2023 *)

def A130233(n): return int(log(3/2 +n*sqrt(5), golden_ratio))
def A130239(n): return A130233(floor(sqrt(n)))
[A130239(n) for n in range(121)] # G. C. Greubel, Mar 18 2023

a(n) = max(k | Fib(k)^2 <= n) = A130233(floor(sqrt(n))).
a(n) = floor(arcsinh(sqrt(5n)/2)/log(phi)), where phi=(1+sqrt(5))/2.
G.f.: (1/(1-x))*Sum_{k>=1} x^(Fib(k)^2).

A130244 Partial sums of the 'upper' Lucas Inverse A130242.

Original entry on oeis.org

0, 0, 0, 2, 5, 9, 13, 17, 22, 27, 32, 37, 43, 49, 55, 61, 67, 73, 79, 86, 93, 100, 107, 114, 121, 128, 135, 142, 149, 156, 164, 172, 180, 188, 196, 204, 212, 220, 228, 236, 244, 252, 260, 268, 276, 284, 292, 300, 309, 318, 327, 336, 345, 354, 363, 372, 381, 390

Offset: 0

Hieronymus Fischer, May 19 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..2500

Other related sequences: A000032, A130241, A130243, A130245, A130246, A130248, A130252, A130258, A130262. Fibonacci inverse see A130233 - A130240, A104162.

[0,0] cat [(&+[Ceiling(Log(k + 1/2)/Log((1+Sqrt(5))/2)) : k in [0..n]]): n in [1..50]]; // G. C. Greubel, Sep 12 2018

Join[{0, 0}, Table[Sum[Ceiling[Log[GoldenRatio, k + 1/2]], {k, 0, n}], {n, 1, 50}]] (* G. C. Greubel, Sep 12 2018 *)

for(n=-1,50, print1(if(n==-1, 0, if(n==0, 0, sum(k=0, n, ceil(log(k + 1/2)/log((1+sqrt(5))/2))))), ", ")) \\ G. C. Greubel, Sep 12 2018

a(n) = Sum_{k=0..n} A130242(k).
a(n) = n*A130242(n) - A000032(A130242(n) +1) for n>=3.
G.f.: x/(1-x)^2*(2*x^2 + Sum{k>=2, x^Lucas(k)}).

A132665 a(1)=1, a(2)=3, a(n) = a(n-1) + n if the minimal positive integer not yet in the sequencer is greater than a(n-1), else a(n) = a(n-1)-1.

Original entry on oeis.org

1, 3, 2, 6, 5, 4, 11, 10, 9, 8, 7, 19, 18, 17, 16, 15, 14, 13, 12, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 87, 86, 85, 84, 83, 82, 81, 80, 79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69

Offset: 1

Hieronymus Fischer, Sep 15 2007

nonn

Also: a(1)=1, a(2)=3, a(n) = maximal positive number < a(n-1) not yet in the sequence, if it exists, else a(n) = a(n-1) + n.
Also: a(1)=1, a(2)=3, a(n) = a(n-1) - 1, if a(n-1) - 1 > 0 and has not been encountered so far, else a(n) = a(n-1) + n.
A permutation of the positive integers. The sequence is self-inverse, in that a(a(n)) = n.

Cf. A000045, A104152, A130233.
For an analog concerning Lucas numbers see A132664.
See A132666-A132674 for sequences with a similar recurrence rule.

G.f.: g(x) = (F'(x) - x^2 - 1/(1-x))/(1-x) where F(x) = Sum_{k>=0} x^Fibonacci(k). F(x) is the g.f. of the Fibonacci indicator sequence (see A104162) and F'(x) = derivative of F(x).
a(n) = Fibonacci(Fibonacci_inverse(n+1) + 2) - n - 3 = A000045(A130233(n+1) + 2) - n - 3.
a(n) = A000032(floor(log_phi(sqrt(5)*(n+1) + 1) + 2)) - n - 3, where phi = (1 + sqrt(5))/2 is the golden ratio.
a(n) = A000032(floor(log_phi(sqrt(5)*n + 2*phi) + 2)) - n - 3.

A130238 Partial sums of A130237.

Original entry on oeis.org

0, 2, 8, 20, 36, 61, 91, 126, 174, 228, 288, 354, 426, 517, 615, 720, 832, 951, 1077, 1210, 1350, 1518, 1694, 1878, 2070, 2270, 2478, 2694, 2918, 3150, 3390, 3638, 3894, 4158, 4464, 4779, 5103, 5436, 5778, 6129, 6489, 6858, 7236, 7623, 8019, 8424, 8838

Offset: 0

Hieronymus Fischer, May 17 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000045, A130233, A130234, A130235, A130236, A130237, A130239, A130240, A130243, A130246, A130248, A130239, A130251, A130253, A130257, A130261.

[(&+[j*Floor(Log(3/2 +j*Sqrt(5))/Log((1+Sqrt(5))/2)): j in [0..n]]): n in [0..70]]; // G. C. Greubel, Mar 18 2023

a[n_]:= a[n]= Sum[j*Floor[Log[GoldenRatio, 3/2 +j*Sqrt[5]]], {j,0,n}];
Table[a[n], {n,0,70}] (* G. C. Greubel, Mar 18 2023 *)

def A130238(n): return sum(j*int(log(3/2 +j*sqrt(5), golden_ratio)) for j in range(n+1))
[A130238(n) for n in range(71)] # G. C. Greubel, Mar 18 2023

a(n) = Sum_{k=0..n} A130237(k).
a(n) = (n*(n+1)*A130233(n) - (Fib(A130233(n)) - 1)*(Fib(A130233(n) + 1) - 1))/2.
G.f.: (1/(1-x)^3)*Sum_{k>=1} (Fib(k)*(1-x) + x)*x^Fib(k).

A130247 Inverse Lucas (A000032) numbers: index k of a Lucas number such that Lucas(k)=n; max(k|Lucas(k) < n), if there is no such index.

Original entry on oeis.org

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

Offset: 1

Hieronymus Fischer, May 19 2007, Jul 02 2007

nonn

Inverse of the Lucas sequence (A000032), since a(Lucas(n))=n for n >= 0 (see A130241 and A130242 for other versions). Same as A130241 except for n=1.

			a(2)=0, since Lucas(0)=2; a(10)=4, since Lucas(4) = 7 < 10 but Lucas(5) = 11 > 10.

G. C. Greubel, Table of n, a(n) for n = 1..10000

For partial sums see A130248. Other related sequences: A000032, A130241, A130242, A130245, A130249, A130255, A130259. Indicator sequence A102460. For Fibonacci inverse see A130233 - A130240, A104162.

Join[{1, 0}, Table[Floor[Log[GoldenRatio, n + 1/2]], {n, 3, 50}]] (* G. C. Greubel, Dec 21 2017 *)

from itertools import islice, count
def A130247_gen(): # generator of terms
    yield from (1,0)
    a, b = 3, 4
    for i in count(2):
        yield from (i,)*(b-a)
        a, b = b, a+b
A130247_list = list(islice(A130247_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n)=c(n), if (n^2-4)/5 is a square number, a(n)=s(n), if (n^2+4)/5 is a square number and a(n)=floor(log_phi(n)) otherwise, where s(n)=floor(arcsinh(n/2)/log(phi)), c(n)=floor(arccosh(n/2)/log(phi)) and phi=(1+sqrt(5))/2.
a(n) = A130241(n) except for n=2.
G.f.: g(x) = (1/(1-x))*(Sum_{k>=1} x^Lucas(k)) - x^2.
a(n) = floor(log_phi(n+1/2)) for n >= 3, where phi is the golden ratio.

cp's OEIS Frontend

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

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A130242 Minimal index k of a Lucas number such that Lucas(k)>=n (the 'upper' Lucas (A000032) Inverse).

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A130259 Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).

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A130261 Partial sums of the 'lower' even Fibonacci Inverse A130259.

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A271354 Products of two distinct Fibonacci numbers, both greater than 1.

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A130239 Maximal index k of the square of a Fibonacci number such that Fib(k)^2 <= n (the 'lower' squared Fibonacci Inverse).

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A130244 Partial sums of the 'upper' Lucas Inverse A130242.

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