A052011 -id:A052011 - cp's OEIS frontend

A052012 Number of primes between successive Lucas numbers.

1, 0, 1, 0, 2, 2, 4, 6, 9, 15, 20, 31, 48, 72, 110, 170, 257, 400, 608, 950, 1448, 2256, 3487, 5413, 8440, 13118, 20478, 31932, 49995, 78222, 122553, 192262, 301826, 474039, 745772, 1173270, 1848000, 2912623, 4593723, 7249438, 11448047

Offset: 1

Patrick De Geest, Nov 15 1999

			Between L(7)=29 and L(8)=47 we find the following primes: 31, 37, 41 and 43 hence a(7)=4.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000032, A000204, A000720, A001606, A005479, A052011, A277062.

a052012 n = a052012_list !! (n-1)
a052012_list = c 1 0 $ tail a000204_list where
  c x y ls'@(l:ls) | x < l     = c (x+1) (y + a010051 x) ls'
                   | otherwise = y : c (x+1) 0 ls
-- Reinhard Zumkeller, Dec 18 2011

PrimePi[Last[#]-1]-PrimePi[First[#]]&/@Partition[LucasL[ Range[45]],2,1] (* Harvey P. Dale, Jun 28 2011 *)

a(n) = pi(L(n + 1) - 1) - pi(L(n)), where pi is the prime counting function (A000720) and L = A000032. - Wesley Ivan Hurt, Nov 09 2023

a(n) = A277062(n+1) - A277062(n) - [n+1 in A001606], 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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A052012 Number of primes between successive Lucas numbers.

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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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