A055498 -id:A055498 - cp's OEIS frontend

A073021 Apart from initial 0, same as A055498.

1, 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

Offset: 0

dead

A362883 a(n) = A055498(n) - A055500(n).

Original entry on oeis.org

-1, 0, 0, 0, 0, 4, 6, 12, 24, 42, 68, 122, 208, 336, 552, 904, 1464, 2378, 3848, 6232, 10090, 16338, 26446, 42802, 69252, 112072, 181332, 293412, 474762, 768190, 1242960, 2011162, 3254150, 5265324, 8519478, 13784866, 22304378, 36089262, 58393658, 94482964, 152876664

Offset: 0

Philip Baciaz, May 07 2023

sign

After the initial zeros, {a(n)} seems approximately linear on a log scale (not a surprise with the prime number theorem in mind?), a(n-1)/a(n) seems to converge to the golden ratio (A001622), and 1/a(5) + 1/a(6) + ... + 1/a(n) seems to converge to 0.60086367622...

Cf. A055500, A055498, A001622.

b:= proc(n, t) option remember; `if`(n<3, n+1, (h-> `if`(t=1,
      prevprime(h), nextprime(h)))(t+b(n-1, t)+b(n-2, t)))
    end:
a:= n-> b(n,-1)-b(n,1):
seq(a(n), n=1..50);  # Alois P. Heinz, May 12 2023

b[n_, t_] := b[n, t] = If[n < 3, n+1, Function[h, If[t == 1, NextPrime[h, -1], NextPrime[h]]][t + b[n-1, t] + b[n-2, t]]];
a[n_] := If[n == 0, -1, b[n-1, -1] - b[n-1, 1]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, May 19 2024, after Alois P. Heinz *)

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

Original entry on oeis.org

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

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

A055499 a(0)=0, a(1)=1, a(n) = smallest prime > a(n-1)+a(n-2).

Original entry on oeis.org

0, 1, 2, 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

Robert G. Wilson v, Table of n, a(n) for n = 0..1000

Cf. A055498, A055500, A055501, A055502.

a[0] = 0; a[1] = 1; a[n_] := a[n] = NextPrime[a[n - 1] + a[n - 2]]; Array[a, 36, 0] (* Robert G. Wilson v, Mar 13 2013 *)

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 *)

A073022 Each term is the smallest prime > the sum of the previous 2 terms.

Original entry on oeis.org

1, 1, 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

Offset: 0

Phil Carmody, Aug 15 2002

nonn

			After 1,1, the next prime >2 is 3.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A055498.

nxt[{a_,b_}]:={b,NextPrime[a+b]}; NestList[nxt,{1,1},40][[All,1]] (* Harvey P. Dale, Apr 14 2022 *)

l=1; h=1; print1("1, 1, "); while(l<2^32,t=l+h+1; while(!isprime(t),t+=1); print1(t,", "); l=h; h=t)

a(n+1)=nextprime(a(n)+a(n-1))

A351140 a(1) = 1, a(n) = smallest prime > a(n-1) + n.

Original entry on oeis.org

1, 5, 11, 17, 23, 31, 41, 53, 67, 79, 97, 113, 127, 149, 167, 191, 211, 233, 257, 281, 307, 331, 359, 389, 419, 449, 479, 509, 541, 577, 613, 647, 683, 719, 757, 797, 839, 881, 929, 971, 1013, 1061, 1109, 1163, 1213, 1277, 1327, 1381, 1433, 1487, 1543, 1597

Offset: 1

Alex Ratushnyak, Feb 02 2022

nonn

The sequence with >= in place of > is essentially the same after the first three terms: 1, 3, 7, 11, 17, 23, 31, ...

			The smallest prime above 1+2 is 5, so a(2)=5.
The smallest prime above 5+3 is 11, so a(3)=11.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A055498, A093503.

R:= 1: p:= 1:
for n from 2 to 100 do
  p:= nextprime(p+n);
  R:= R,p;
od:
R; # Robert Israel, Nov 20 2023

a[1] = 1; a[n_] := a[n] = NextPrime[a[n - 1] + n]; Array[a, 50] (* Amiram Eldar, Feb 03 2022 *)

lista(nn) = my(list = List(), last = 1); listput(list, last); for (n=2, nn, last = nextprime(last + n +1); listput(list, last);); Vec(list); \\ Michel Marcus, Feb 03 2022

from sympy import nextprime
p = 1
for i in range(2,1000):
  print(p, end=",")
  p = nextprime(p+i)

cp's OEIS Frontend

A073021 Apart from initial 0, same as A055498.

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A362883 a(n) = A055498(n) - A055500(n).

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A055500 a(0)=1, a(1)=1, a(n) = largest 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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A055499 a(0)=0, a(1)=1, 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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Mathematica

A073022 Each term is the smallest prime > the sum of the previous 2 terms.

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A351140 a(1) = 1, a(n) = smallest prime > a(n-1) + n.

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