A055500 -id:A055500 - cp's OEIS frontend

A065436 Duplicate of A055500.

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

Offset: 1

dead

A068523 Essentially the same as A055500.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 23, 37, 59, 89, 139, 227, 359, 577, 929, 1499, 2423, 3919, 6337, 10253

Offset: 1

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 *)

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

A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

Original entry on oeis.org

41, 83, 197, 199, 223, 240, 251, 281, 287, 340, 371, 401, 439, 491, 510, 593, 660, 733, 803, 857, 864, 883, 931, 941, 961, 983, 990, 991, 1012, 1060, 1061, 1099, 1104, 1187, 1236, 1283, 1313, 1361, 1381, 1392, 1433, 1439, 1493, 1511, 1523, 1524, 1553

Offset: 1

Jud McCranie, May 30 2000

nonn

			41 can be expressed as 41 or 11+13+17 or 2+3+5+7+11+13, so 41 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Robert Israel, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054999, A055500, A055001.

N:= 10^4: # to get all terms <= N
P:= [0,op(select(isprime, [2,seq(i,i=3..N,2)]))]:
nP:= nops(P);
S:= ListTools:-PartialSums(P):
V:= Vector(N):
for i from 1 to nP-1 do
  for j from i+1 to nP while S[j] - S[i] <= N do
     V[S[j]-S[i]]:= V[S[j]-S[i]]+1
od od:
select(t -> V[t] = 3, [$1..N]): # Robert Israel, Apr 05 2017

Module[{nn = 300, s}, s = Array[Prime, nn]; Keys@ Take[Select[KeySort@ Merge[Table[PositionIndex@ Map[Total, Partition[s, k, 1]], {k, nn/2}], Identity], Length@ # == 3 &], Floor[nn/6]]] (* Michael De Vlieger, Apr 06 2017, Version 10 *)

A054845(a(n)) = 3. - Ray Chandler, Sep 20 2023

A054999 Integers that can be expressed as the sum of consecutive primes in exactly 4 ways.

Original entry on oeis.org

1151, 1164, 1320, 1367, 1650, 1854, 1951, 2393, 2647, 2689, 2856, 2867, 3198, 3264, 3389, 3754, 4200, 4920, 4957, 5059, 5100, 5153, 5770, 5999, 6504, 7451, 7901, 8152, 8819, 10134, 10320, 10499, 10536, 10649, 10859, 10949, 11058, 12294

Offset: 1

Jud McCranie, May 30 2000

nonn

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054997, A054998, A055500, A055001.

A054845(a(n)) = 4. - Ray Chandler, Sep 20 2023

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

A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

Original entry on oeis.org

5, 17, 23, 31, 36, 53, 59, 60, 67, 71, 72, 90, 97, 100, 101, 109, 112, 119, 120, 127, 131, 138, 139, 143, 152, 173, 180, 181, 187, 204, 210, 211, 221, 228, 233, 258, 263, 269, 271, 276, 300, 304, 323, 330, 331, 349, 353, 372, 373, 379, 384, 390, 395, 408

Offset: 1

Jud McCranie, May 30 2000

nonn

			5 can be expressed as 5 or 2+3, so 5 is in the sequence.

R. K. Guy, Unsolved Problems in Number Theory, section C2.

Ray Chandler, Table of n, a(n) for n = 1..10000
Carlos Rivera, Puzzle 46. Primes expressible as sum of consecutive primes in K ways, The Prime Puzzles and Problems Connection.

Cf. A054845, A054859, A054996, A054998, A054999, A055500, A055001.

A054845(a(n)) = 2. - Ray Chandler, Sep 20 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 *)

cp's OEIS Frontend

A065436 Duplicate of A055500.

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A068523 Essentially the same as A055500.

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

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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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A054998 Integers that can be expressed as the sum of consecutive primes in exactly 3 ways.

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A054999 Integers that can be expressed as the sum of consecutive primes in exactly 4 ways.

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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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A054997 Integers that can be expressed as the sum of consecutive primes in exactly 2 ways.

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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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