A052012 -id:A052012 - cp's OEIS frontend

A277062 Number of primes <= n-th Lucas number.

1, 0, 2, 2, 4, 5, 7, 10, 15, 21, 30, 46, 66, 98, 146, 218, 329, 500, 757, 1158, 1766, 2716, 4164, 6420, 9907, 15320, 23760, 36878, 57356, 89288, 139283, 217506, 340059, 532321, 834147, 1308186, 2053958, 3227229, 5075229, 7987852, 12581575, 19831014

Offset: 0

Vincenzo Librandi, Nov 09 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..101 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000032, A000720, A001606, A005479, A052012, A054782.

Magma
```
[#PrimesUpTo(Lucas(n)): n in [0..41]];
```

a:= n-> numtheory[pi]((<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2016

Mathematica
```
Table[PrimePi[LucasL[n]], {n, 0, 50}]
```

a(n) = A000720(A000032(n)). - Michel Marcus, Jun 10 2024

A052011 Number of primes between successive Fibonacci numbers exclusive.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 16, 23, 37, 55, 84, 125, 198, 297, 458, 704, 1087, 1673, 2602, 4029, 6263, 9738, 15186, 23704, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419527, 13298630

Offset: 1

Patrick De Geest, Nov 15 1999

With the given sequence data, we see that neither endpoint is included, so we count primes p in the open interval F(n)Jeppe Stig Nielsen, Jun 06 2015

			Between Fib(9)=34 and Fib(10)=55 we find the following primes: 37, 41, 43, 47 and 53 hence a(9)=5.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated from the data at A054782 and A001605)

Cf. A000040, A001605, A005478 (endpoint primes), A010051, A052012, A054782.

a052011 n = a052011_list !! (n-1)
a052011_list = c 0 0 $ drop 2 a000045_list where
  c x y fs'@(f:fs) | x < f     = c (x+1) (y + a010051 x) fs'
                   | otherwise = y : c (x+1) 0 fs
-- Reinhard Zumkeller, Dec 18 2011

for n from 1 to 43 do T[n]:= numtheory:-pi(combinat:-fibonacci(n)) od:
seq(T[n]-T[n-1]-`if`(isprime(combinat:-fibonacci(n)),1,0), n=2..43); # Robert Israel, Jun 08 2015

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n]+1,Fibonacci[n+1]-1}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
pbf[n_]:=Module[{fib1=If[PrimeQ[Fibonacci[n+1]],PrimePi[Fibonacci[n+1]-1], PrimePi[ Fibonacci[n+1]]], fib0=If[PrimeQ[Fibonacci[n]], PrimePi[ Fibonacci[n]+1],PrimePi[Fibonacci[n]]]},Max[0,fib1-fib0]]; Array[pbf,50] (* Harvey P. Dale, Mar 01 2012 *)

a(n)=my(s); forprime(p=fibonacci(n)+1,fibonacci(n+1)-1,s++); s \\ Charles R Greathouse IV, Jun 08 2015

a(n) = pi(F(n+1)-1) - pi(F(n)) = A000720(A000045(n+1)-1) - A000720(A000045(n)). - Jonathan Vos Post, Mar 08 2010; corrected by Jeppe Stig Nielsen, Jun 06 2015
a(n) ~ phi^(n-1)/(n*sqrt(5)*log(phi)), where phi = (1+sqrt(5))/2 is the golden ratio. - Charles R Greathouse IV, Jun 08 2015
a(n) = A054782(n+1) - A054782(n) - [n+1 in A001605], where [] denotes the Iverson bracket. - Amiram Eldar, Jun 10 2024

A131354 Number of primes in the open interval between successive tribonacci numbers.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 3, 5, 8, 12, 23, 38, 61, 109, 179, 312, 537, 920, 1598, 2779, 4835, 8461, 14784, 25984, 45696, 80505, 142165, 251300, 444930, 788828, 1400756, 2489594, 4430712, 7892037, 14073786, 25118167, 44869652, 80223172, 143535369, 257014148, 460524864, 825732764

Offset: 0

Jonathan Vos Post, Oct 21 2007

nonn

This is to tribonacci numbers A000073 as A052011 is to Fibonacci numbers and as A052012 is to Lucas numbers A000204. It is mere coincidence that all values until a(12) = 38 are themselves Fibonacci. The formula plus the known asymptotic prime distribution gives the asymptotic approximation of this sequence, which is the same even if we use one of the alternative definitions of tribonacci with different initial values.

			Between Trib(8)=24 and Trib(9)=44 we find the following primes: 29, 31, 37, 41, 43, hence a(8)=5.

Cf. A000073, A000720, A092836, A052011, A052012, A056816.

A131354 := proc(n)
    a := 0 ;
    for k from 1+A000073(n)  to A000073(n+1)-1 do
        if isprime(k) then
            a := a+1 ;
        end if;
    end do;
    a ;
end proc: # R. J. Mathar, Dec 14 2011

trib[n_] := SeriesCoefficient[x^2/(1 - x - x^2 - x^3), {x, 0, n}];
a[n_] := PrimePi[trib[n + 1] - 1] - PrimePi[trib[n]];
a /@ Range[0, 42] (* Jean-François Alcover, Apr 10 2020 *)

\\ here b(n) is A000073(n).
b(n)={polcoef(x^2/(1-x-x^2-x^3) + O(x*x^n), n)}
a(n)={primepi(b(n+1)-1) - primepi(b(n))} \\ Andrew Howroyd, Jan 02 2020

a(n) = A000720(A000073(n+1) - 1) - A000720(A000073(n)) for n >= 3. [formula edited Andrew Howroyd, Jan 02 2020]

Terms a(26) and beyond from Andrew Howroyd, Jan 02 2020

A130973 Number of primes between successive pairs of twin primes, for a(n) > 0.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 4, 2, 1, 3, 1, 2, 3, 10, 4, 7, 4, 3, 2, 1, 2, 18, 2, 2, 17, 1, 2, 6, 9, 3, 1, 1, 1, 8, 3, 2, 15, 1, 4, 1, 1, 7, 7, 4, 4, 3, 4, 1, 1, 7, 2, 5, 1, 5, 18, 2, 5, 4, 3, 1, 5, 1, 18, 12, 2, 8, 1, 4, 2, 5, 4, 1, 1, 1, 9, 10

Offset: 1

Omar E. Pol, Aug 23 2007

a(k) corresponds to the k-th term in the isolated prime sequence A007510 or A134797. a(1) corresponds to 23. a(2) corresponds to 37. a(3) corresponds to 47 and 53. - Enrique Navarrete, Jan 28 2017

Lengths of the runs of consecutive integers in A176656. - R. J. Mathar, Feb 19 2017

C. K. Caldwell, The Prime Pages.
Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Cf. A001223, A007510 (isolated primes), A027883, A048614, A048198, A052011, A052012, A061273, A076777, A073784, A082602, A088700, A179067 (clusters of twin primes).

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A277062 Number of primes <= n-th Lucas number.

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A052011 Number of primes between successive Fibonacci numbers exclusive.

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A131354 Number of primes in the open interval between successive tribonacci numbers.

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A130973 Number of primes between successive pairs of twin primes, for a(n) > 0.

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