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

Author

Patrick De Geest, Nov 15 1999

Keywords

• nonn
• nice

Examples

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

Links

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

Crossrefs

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

Programs

Haskell

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

Mathematica

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

Formula

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