A055500 a(0)=1, a(1)=1, a(n) = largest prime <= a(n-1) + a(n-2).

1, 1, 2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 359, 577, 929, 1499, 2423, 3919, 6337, 10253, 16573, 26821, 43391, 70207, 113591, 183797, 297377, 481171, 778541, 1259701, 2038217, 3297913, 5336129, 8633983, 13970093, 22604069, 36574151, 59178199, 95752333

Offset: 0

Author

N. J. A. Sloane, Jul 08 2000

Keywords

• nonn
• easy
• nice

Comments

Or might be called Ishikawa primes, as he proved that prime(n+2) < prime(n) + prime(n+1) for n > 1. This improves on Bertrand's Postulate (Chebyshev's theorem), which says prime(n+2) < prime(n+1) + prime(n+1). - Jonathan Sondow, Sep 21 2013

Examples

			a(8) = 23 because 23 is largest prime <= a(7) + a(6) = 17 + 11 = 28.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1000 (first 100 terms from Zak Seidov)
• Heihachiro Ishikawa, Über die Verteilung der Primzahlen, Sci. Rep. Tokyo Bunrika Daigaku, Sect. A 2 (1934), 27-40.

Crossrefs

Cf. A007917, A055498, A055499, A055400, A055401, A055502, A065435.

Programs

Haskell

a055500 n = a055500_list !! n
a055500_list = 1 : 1 : map a007917
               (zipWith (+) a055500_list $ tail a055500_list)
-- Reinhard Zumkeller, May 01 2013

Mathematica

PrevPrim[n_] := Block[ {k = n}, While[ !PrimeQ[k], k-- ]; Return[k]]; a[1] = a[2] = 1; a[n_] := a[n] = PrevPrim[ a[n - 1] + a[n - 2]]; Table[ a[n], {n, 1, 42} ]
(* Or, if version >= 6 : *)a[0] = a[1] = 1; a[n_] := a[n] = NextPrime[ a[n-1] + a[n-2] + 1, -1]; Table[a[n], {n, 0, 100}](* Jean-François Alcover, Jan 12 2012 *)
nxt[{a_,b_}]:={b,NextPrime[a+b+1,-1]}; Transpose[NestList[nxt,{1,1},40]] [[1]] (* Harvey P. Dale, Jul 15 2013 *)

Python

from sympy import prevprime; L = [1, 1]
for _ in range(36): L.append(prevprime(L[-2] + L[-1] + 1))
print(*L, sep = ", ")  # Ya-Ping Lu, May 05 2023

Formula

a(n) is asymptotic to C*phi^n where phi = (1+sqrt(5))/2 and C = 0.41845009129953131631777132510164822489... - Benoit Cloitre, Apr 21 2003

a(n) = A007917(a(n-1) + a(n-2)) for n > 1. - Reinhard Zumkeller, May 01 2013

a(n) >= prime(n-1) for n > 1, by Ishikawa's theorem. - Jonathan Sondow, Sep 21 2013