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

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

Author

Jonathan Vos Post, Oct 21 2007

Keywords

• nonn

Comments

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.

Examples

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

Crossrefs

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

Programs

Maple

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

Mathematica

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 *)

PARI

\\ 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

Formula

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

Extensions

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