A055499 -id:A055499 - cp's OEIS frontend

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

N. J. A. Sloane, Jul 08 2000

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

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

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.

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

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

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 *)

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

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

A055498 a(0)=0, a(1)=1, a(n) = smallest prime >= a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 2, 3, 5, 11, 17, 29, 47, 79, 127, 211, 347, 563, 911, 1481, 2393, 3877, 6271, 10151, 16427, 26591, 43019, 69623, 112643, 182279, 294923, 477209, 772139, 1249361, 2021501, 3270863, 5292367, 8563237, 13855607, 22418849, 36274471, 58693331, 94967809, 153661163

Offset: 0

N. J. A. Sloane, Jul 08 2000

nonn

			After 3, 5, the next prime >=8 is 11.

Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 0..1001 (first 101 terms from Zak Seidov)

Cf. A073021, A073022, A055499, A055500, A055501, A055502.

Cf. A007918.

a055498 n = a055498_list !! n
a055498_list = 0 : 1 : map a007918
    (zipWith (+) a055498_list $ tail a055498_list)
-- Reinhard Zumkeller, Nov 13 2014

a[0] = 0; a[1] = 1; a[n_] := a[n] = NextPrime[a[n - 1] + a[n - 2] -1]; Array[a, 37, 0] (* Robert G. Wilson v, Mar 13 2013 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==NextPrime[a[n-1]+a[n-2]-1]},a,{n,50}] (* Harvey P. Dale, May 08 2013 *)

a(n)=local(v);if(n<2,n>=0,n++;v=vector(n,i,1);for(i=3,n,v[i]=nextprime(v[i-1]+v[i-2]));v[n]) /* Michael Somos, Feb 01 2004 */

a(n+1) = nextprime(a(n) + a(n-1)) where nextprime(n) is smallest prime >= n.
a(n) is asymptotic to c*phi^n where phi = (1 + sqrt(5))/2 and c = 1.086541275044988562375... - Benoit Cloitre, May 02 2004
a(n) = A055499(n-1) for n>3. - Robert G. Wilson v, Mar 13 2013
a(n) = A007918(a(n-1) + a(n-2)) for n > 1. - Reinhard Zumkeller, Nov 13 2014

A055502 a(0)=0, a(1)=2, a(n) = smallest prime > a(n-1)+a(n-2).

Original entry on oeis.org

0, 2, 3, 7, 11, 19, 31, 53, 89, 149, 239, 389, 631, 1021, 1657, 2683, 4349, 7039, 11393, 18433, 29833, 48271, 78121, 126397, 204521, 330943, 535481, 866431, 1401937, 2268377, 3670319, 5938711, 9609031, 15547769, 25156811, 40704589, 65861461, 106566059, 172427531

Offset: 0

N. J. A. Sloane, Jul 09 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A055498, A055499, A055500, A055501.

A055502 := proc(n) option remember; if n<=0 then n else nextprime(A055502(n-1)+A055502(n-2)); fi; end;

a[0] = 0; a[1] = 2; a[n_] := a[n] = NextPrime[a[n-1] + a[n-2]]; Array[a, 40, 0] (* Amiram Eldar, Sep 24 2023 *)

More terms from Amiram Eldar, Sep 24 2023

A055501 a(0)=1, a(1)=2, a(n) = largest prime < a(n-1)+a(n-2).

Original entry on oeis.org

1, 2, 2, 3, 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

N. J. A. Sloane, Jul 08 2000

nonn

Except for initial terms, same as A055500. - Franklin T. Adams-Watters, Jul 11 2006

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A055498, A055499, A055500, A055502.

Transpose[NestList[{#[[2]],NextPrime[Total[#],-1]}&,{1,2},40]][[1]] (* Harvey P. Dale, May 29 2013 *)

cp's OEIS Frontend

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

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A055498 a(0)=0, a(1)=1, a(n) = smallest prime >= a(n-1) + a(n-2).

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A055502 a(0)=0, a(1)=2, a(n) = smallest prime > a(n-1)+a(n-2).

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A055501 a(0)=1, a(1)=2, a(n) = largest prime < a(n-1)+a(n-2).

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