A104160 -id:A104160 - cp's OEIS frontend

A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci Inverse).

1, 2, 2, 2, 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, 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, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 1

Hieronymus Fischer, May 24 2007, Jul 02 2007

nonn

Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=0 (see A130256 for another version). a(n)+1 is the number of odd Fibonacci numbers (A001519) <= n (for n >= 1).

			a(10)=3 because A001519(3) = 5 <= 10, but A001519(4) = 13 > 10.

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

Cf. partial sums A130257. Other related sequences: A000045, A130233, A130237, A130239, A130256, A130259, A104160. Lucas inverse: A130241 - A130248.

phi:=(1+Sqrt(5))/2; [Floor((1 +Argsinh(Sqrt(5)*n/2)/Log(phi))/2): n in [1..100]]; // G. C. Greubel, Sep 09 2018

Table[Floor[(1 +ArcSinh[Sqrt[5]*n/2]/Log[GoldenRatio])/2], {n, 1, 100}] (* G. C. Greubel, Sep 09 2018 *)

phi=(1+sqrt(5))/2; vector(100, n, floor((1 +asinh(sqrt(5)*n/2)/log(phi))/2)) \\ G. C. Greubel, Sep 09 2018

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

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.

A376013 Prime numbers of the form 3p+8 where p, p+2 and p+6 are prime numbers.

Original entry on oeis.org

23, 41, 59, 131, 311, 941, 1049, 1931, 2579, 3911, 4289, 4451, 6719, 8069, 10391, 10589, 12011, 14369, 26591, 31379, 33521, 35339, 41081, 43889, 58271, 59981, 63059, 64679, 66821, 74759, 77999, 78791, 80051, 80141, 83219, 87071, 94541, 96179

Offset: 1

Zak Seidov, Sep 06 2024

nonn

An integer n is in this list if it is a prime number and (n-8)/3, (n-2)/3, (n+10)/3 are all prime numbers. 23 is a term because it is prime and 5, 7 and 11 are prime numbers.

			5 + 7 + 11 = 23;
11 + 13 + 17 = 41;
17 + 19 + 23 = 59;
41 + 43 + 47 = 131;
101 + 103 + 107 = 311;
311 + 313 + 317 = 941;
347 + 349 + 353 = 1049;
...

Daniel Mondot, Table of n, a(n) for n = 1..10000

Cf. A162001, A104160.

Select[Total /@ Select[Partition[Prime[Range[3500]], 3, 1], Differences[#] == {2, 4} &], PrimeQ] (* Amiram Eldar, Sep 06 2024 *)

list(lim)=my(v=List(),p=5,q=7,s); forprime(r=11,(lim+10)\3, if(r-p==6 && q-p==2 && isprime(s=3*p+8), listput(v,s)); p=q; q=r); Vec(v) \\ Charles R Greathouse IV, Sep 18 2024

a(n) = 3*A162001(n) + 8. - Daniel Mondot, Sep 06 2024
a(n) == 5 (mod 6). - Hugo Pfoertner, Sep 06 2024
a(n) >> n log^4 n. - Charles R Greathouse IV, Sep 18 2024

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A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci 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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A376013 Prime numbers of the form 3p+8 where p, p+2 and p+6 are prime numbers.

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