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 A131354 Number of primes in the open interval between successive tribonacci numbers. 1
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 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. LINKS FORMULA a(n) = A000720(A000073(n+1) - 1) - A000720(A000073(n)) for n >= 3. [formula edited Andrew Howroyd, Jan 02 2020] EXAMPLE Between Trib(8)=24 and Trib(9)=44 we find the following primes: 29, 31, 37, 41, 43, hence a(8)=5. 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 *) PROG (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 CROSSREFS Cf. A000073, A000720, A092836, A052011, A052012, A056816. Sequence in context: A147880 A276527 A020643 * A092360 A129141 A289013 Adjacent sequences: A131351 A131352 A131353 * A131355 A131356 A131357 KEYWORD nonn AUTHOR Jonathan Vos Post, Oct 21 2007 EXTENSIONS Terms a(26) and beyond from Andrew Howroyd, Jan 02 2020 STATUS approved

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Last modified November 27 01:14 EST 2022. Contains 358362 sequences. (Running on oeis4.)