A055502 -id:A055502 - cp's OEIS frontend

A113823 Tribonacci analog of A055502.

0, 2, 3, 7, 13, 29, 53, 97, 181, 337, 617, 1151, 2111, 3881, 7151, 13147, 24181, 44483, 81817, 150497, 276817, 509137, 936469, 1722431, 3168097, 5827001, 10717561, 19712669, 36257237, 66687469, 122657377, 225602099, 414946951, 763206467, 1403755531, 2581909003

Offset: 0

Jonathan Vos Post, Jan 23 2006

This is to the tribonacci sequence as A055502 is to the Fibonacci sequence (i.e. least prime greater than the sum of the previous 2 terms in A055502, least prime greater than the sum of the previous 3 terms in this sequence).

The first 9 positive terms are also elements of A089189 but that coincidence breaks down as a(10) = 617 is a prime p, but p-1 = 616 = 2^3 * 7 * 11 is not cubefree.

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

Cf. A055502, A089189.

a[0] = 0; a[1] = 2; a[2] = 3; a[n_] := a[n] = NextPrime[a[n-1] + a[n-2] + a[n-3]]; Array[a, 40, 0] (* Amiram Eldar, Sep 24 2023 *)

a(0) = 0, a(1) = 2, for n>2: a(n) = smallest prime > a(n-1)+a(n-2)+a(n-3).

More terms from Amiram Eldar, Sep 24 2023

A113884 Pentanacci analog of A055502.

Original entry on oeis.org

0, 2, 3, 7, 13, 29, 59, 113, 223, 439, 877, 1721, 3389, 6653, 13093, 25741, 50599, 99487, 195581, 384509, 755959

Offset: 0

Jonathan Vos Post, Jan 27 2006

This is to the pentanacci sequence as A055502 is to the Fibonacci sequence and A113823 is to the tribonacci sequence (i.e., least prime greater than the sum of the previous 2 terms in A055502, least prime greater than the sum of the previous 3 terms in A113823, least prime greater than the sum of the previous 5 terms in this sequence).

			a(6) = 59 because a(1)+a(2)+a(3)+a(4)+a(5) = 2+3+7+13+29 = 54, the smallest prime beyond 54 is 59.
a(10) = 877 because a(5)+a(6)+a(7)+a(8)+a(9) = 29 + 59 + 113 + 223 + 439 = 863 is prime, the next prime being 14 more, namely 877.

Cf. A055502, A113823.

a(-n) = a(0) = 0, a(1) = 2, for n>1: a(n) = smallest prime > a(n-1)+a(n-2)+a(n-3)+a(n-4)+a(n-5).

A113843 Tetranacci analog of A055502.

Original entry on oeis.org

0, 2, 3, 7, 13, 29, 53, 103, 199, 389, 751, 1447, 2789, 5381, 10369, 19991, 38543, 74287, 143197, 276019, 532061, 1025579, 1976857, 3810517, 7345031, 14158009, 27290429, 52604017, 101397487, 195449957, 376741891, 726193373, 1399782719, 2698167947, 5200885961

Offset: 0

Jonathan Vos Post, Jan 24 2006

This is to the tribonacci sequence as A055502 is to the Fibonacci sequence and A113823 is to the tribonacci sequence (i.e., least prime greater than the sum of the previous 2 terms in A055502, least prime greater than the sum of the previous 3 terms in A113823, least prime greater than the sum of the previous 4 terms in this sequence).

			a(15) = 19991 because a(11)+a(12)+a(13)+a(14) = 1447 + 2789 + 5381 + 10369 = 19986 and 19991 is the smallest prime > 19986.

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

Cf. A055502, A113823.

nxt[{a_,b_,c_,d_}]:={b,c,d,NextPrime[a+b+c+d]}; Transpose[ NestList[ nxt,{0,2,3,7},40]][[1]] (* Harvey P. Dale, Sep 18 2013 *)

a(-n) = a(0) = 0, a(1) = 2, for n>1: a(n) = smallest prime > a(n-1)+a(n-2)+a(n-3)+a(n-4).

More terms from Harvey P. Dale, Sep 18 2013

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

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

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 *)

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A113823 Tribonacci analog of A055502.

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A113884 Pentanacci analog of A055502.

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A113843 Tetranacci analog of A055502.

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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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A055498 a(0)=0, a(1)=1, 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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