A229608 -id:A229608 - cp's OEIS frontend

A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1).

2, 5, 11, 23, 47, 97, 197, 397, 797, 1597, 3203, 6421, 12853, 25717, 51437, 102877, 205759, 411527, 823117, 1646237, 3292489, 6584983, 13169977, 26339969, 52679969, 105359939, 210719881, 421439783, 842879579, 1685759167, 3371518343

Offset: 1

N. J. A. Sloane, Jul 07 2000

nonn

It appears that lim_{n->infinity} a(n)/2^n exists and is approximately 1.569985585.... - Franklin T. Adams-Watters, Nov 11 2011
This is a B_2 sequence. - Thomas Ordowski, Sep 23 2014 See the link.
Conjecture: lim_{n->infinity} a(n)/A006992(n) = 5.1648264... - Thomas Ordowski, Apr 05 2015

Zak Seidov and Michael De Vlieger, Table of n, a(n) for n = 1..1000 (First 100 terms from Zak Seidov)
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), #12.5.4.
Eric Weisstein's World of Mathematics, B_2-Sequence.

Cf. A006992, A060264, A163963.
Values of a(n)-2*a(n-1) in A163469. - Zak Seidov, Jul 28 2009
Cf. A065545 (with a(1)=3). - Zak Seidov, Feb 04 2016
Row 1 of A229608.

A055496 := proc(n) option remember; if n=1 then 2 else nextprime(2*A055496(n-1)); fi; end;

NextPrim[n_Integer] := Block[ {k = n + 1}, While[ !PrimeQ[k], k++ ]; Return[k]]; a[1] = 2; a[n_] := NextPrim[ 2*a[n - 1]]; Table[ a[n], {n, 1, 31} ]
a[1]=2;a[n_]:=a[n]=Prime[PrimePi[2*a[n-1]]+1];Table[a[n],{n,40}] (* Zak Seidov, Feb 16 2006 *)
NestList[ NextPrime[2*# ]&,2,100] (* Zak Seidov, Jul 28 2009 *)

print1(a=2);for(n=2,20,print1(", ",a=nextprime(a+a))) \\ Charles R Greathouse IV, Jul 19 2011

a(n+1) = A060264(a(n)). - Peter Munn, Oct 23 2017

Mathematica updated by Jean-François Alcover, Jun 19 2013

A194598 Union of A080359 and A164294.

Original entry on oeis.org

2, 3, 13, 19, 31, 43, 53, 61, 71, 73, 101, 103, 109, 113, 131, 139, 151, 157, 173, 181, 191, 193, 199, 229, 233, 239, 241, 251, 269, 271, 283, 293, 311, 313, 349, 353, 373, 379, 409, 419, 421, 433, 439, 443, 463, 491, 499, 509, 523, 571, 577, 593, 599, 601

Offset: 1

Vladimir Shevelev, Aug 30 2011

nonn

Every greater of twin primes (A006512), beginning with 13, is in the sequence.
A very simple sieve for the generation of the terms is the following: Let p_n be the n-th prime. Consider consecutive intervals of the form (2p_n, 2p_{n+1}), n=1,2,... From every interval containing at least one prime we take the first one and remove it from the set of all primes. Then all remaining primes form the sequence. Let us demonstrate this sieve: For primes 2,3,5,7,11,... consider intervals (4,6), (6,10), (10,14), (14,22), (22,26), (26,34), ... . Removing from the set of all primes the first prime of each interval, i.e., 5,7,11,17,23,29,... ,we obtain 2,3,13,19,31, etc.
This sequence and A164368 are the mutually wrapping up sequences:
   a(1) <= A164368(1) <= a(2) <= A164368(2) <= ...
Following the steps to generate T(n,1) in A229608 provides an alternate method of generating this sequence. - Bob Selcoe, Oct 27 2015

Alois P. Heinz, Table of n, a(n) for n = 1..1000
V. Shevelev, Ramanujan and Labos primes, their generalizations and classifications of primes, arXiv:0909.0715 [math.NT], 2009, 2011.
V. Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012) Article 12.5.4
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.
J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, J. Integer Seq. 14 (2011) Article 11.6.2.

Cf. A080359, A164294, A164368, A193507, A194186, A212493, A212541, A229608.

If the first two terms are omitted we get A164333.

primePiMax = 200;
Join[{2, 3}, Select[Table[{(Prime[k-1] + 1)/2, (Prime[k] - 1)/2}, {k, 3, primePiMax}], AllTrue[Range[#[[1]], #[[2]]], CompositeQ]&][[All, 2]]*2+1] (* Jean-François Alcover, Aug 18 2018 *)

First column of array A229608. - Bob Selcoe, Oct 27 2015

For n >= 3, a(n) = A164333(n-2). - Peter Munn, Aug 30 2017

A229607 Square array read by antidiagonals downwards in which each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 2*p.

Original entry on oeis.org

2, 3, 11, 5, 19, 17, 7, 37, 31, 29, 13, 73, 61, 53, 41, 23, 139, 113, 103, 79, 47, 43, 277, 223, 199, 157, 89, 59, 83, 547, 443, 397, 313, 173, 113, 67, 163, 1093, 883, 787, 619, 337, 223, 131, 71, 317, 2179, 1759, 1571, 1237, 673, 443, 257, 139, 97, 631

Offset: 1

Clark Kimberling, Sep 26 2013

Conjectures: (row 1) = A006992, (column 1) = A104272, and for each row r(k), the limit of r(k)/2^k exists. For rows 1 to 4, the respective limits are 0.303976..., 4.249137..., 6.857407..., 12.235210... .
From Pontus von Brömssen, Jan 18 2025: (Start)
Regarding the conjectures above:
  - Row 1 is A006992 by definition.
  - Column 1 is A164368, not A104272. It seems that the first column would be A104272 if no duplicates were allowed, i.e., if the prime p in a row were followed by the largest prime < 2*p not in a previous row; see A380277.
  - The existence of the limits should follow from a strong version of Bertrand's postulate. For row 1, see formula in A006992.
(End)

			Northwest corner:
   2,    3,    5,    7,   13,   23,   43,   83, ...
  11,   19,   37,   73,  139,  277,  547, 1093, ...
  17,   31,   61,  113,  223,  443,  883, 1759, ...
  29,   53,  103,  199,  397,  787, 1571, 3137, ...
  41,   79,  157,  313,  619, 1237, 2473, 4943, ...
  47,   89,  173,  337,  673, 1327, 2647, 5281, ...

Pontus von Brömssen, Table of n, a(n) for n = 1..5050 (first 100 antidiagonals)
Wikipedia, Bertrand's postulate.

Cf. A006992, A104272, A164368, A229608, A229609, A229610, A380277.

seqL = 14; arr1[1] = {2}; Do[AppendTo[arr1[1], NextPrime[2*Last[arr1[1]], -1]], {seqL}]; Do[tmp = Union[Flatten[Map[arr1, Range[z]]]]; arr1[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr1[z], NextPrime[2*Last[arr1[z]], -1]], {seqL}], {z, 2, 12}]; m = Map[arr1, Range[12]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Incorrect comment deleted by Peter Munn, Aug 15 2017

A229609 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 3*p.

Original entry on oeis.org

2, 5, 3, 13, 7, 11, 37, 19, 31, 17, 109, 53, 89, 47, 23, 317, 157, 263, 139, 67, 29, 947, 467, 787, 409, 199, 83, 41, 2837, 1399, 2357, 1223, 593, 241, 113, 43, 8501, 4177, 7069, 3659, 1777, 719, 337, 127, 59, 25471, 12527, 21193, 10973, 5323, 2153, 1009

Offset: 1

Clark Kimberling, Sep 26 2013

Conjectures: (row 1) = A126031, (column 1) = A164952, and for each row r(k), the limit of r(k)/3^k exists. For rows 1 to 4, the respective limits are 0.431270..., 0.636059..., 3.229697..., 5.015914... .

			Northwest corner:
   2,  5,  13,  37,  109,  317, ...
   3,  7,  19,  53,  157,  467, ...
  11, 31,  89, 263,  787, 2357, ...
  17, 47, 139, 409, 1223, 3659, ...
  23, 67, 199, 593, 1777, 5323, ...
  29, 83, 241, 719, 2153, 6451, ...

Cf. A126031, A164952, A229607, A229608, A229610.

seqL = 14; arr1[1] = {2}; Do[AppendTo[arr1[1], NextPrime[3*Last[arr1[1]], -1]], {seqL}];  Do[tmp = Union[Flatten[Map[arr1, Range[z]]]]; arr1[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr1[z], NextPrime[3*Last[arr1[z]], -1]], {seqL}], {z, 2, 22}]; m = Map[arr1, Range[22]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Incorrect comment deleted by Peter Munn, Aug 15 2017

A229610 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the least prime > 3*p.

Original entry on oeis.org

2, 7, 3, 23, 11, 5, 71, 37, 17, 13, 223, 113, 53, 41, 19, 673, 347, 163, 127, 59, 29, 2027, 1049, 491, 383, 179, 89, 31, 6089, 3163, 1481, 1151, 541, 269, 97, 43, 18269, 9491, 4447, 3457, 1627, 809, 293, 131, 47, 54829, 28477, 13367, 10391, 4889, 2437, 881

Offset: 1

Clark Kimberling, Sep 26 2013

Conjectures: (row 1) = A076656, (column 1) = A164958, and for each row r(k), the limit of r(k)/3^k exists. For rows 1 to 4, the respective limits are 0.928655..., 1.447047..., 2.038260..., 4.753271... .

			Northwest corner:
   2,  7,  23,  71,  223,  673, ...
   3, 11,  37, 113,  347, 1049, ...
   5, 17,  53, 163,  491, 1481, ...
  13, 41, 127, 383, 1151, 3457, ...
  19, 59, 179, 541, 1627, 4889, ...
  29, 89, 269, 809, 2437, 7331, ...

Cf. A076656, A164958, A229607, A229608, A229609.

seqL = 14; arr2[1] = {2}; Do[AppendTo[arr2[1], NextPrime[3*Last[arr2[1]]]], {seqL}]; Do[tmp = Union[Flatten[Map[arr2, Range[z]]]]; arr2[z] = {Prime[NestWhile[# + 1 &, 1, PrimePi[tmp[[#]]] - # == 0 &]]}; Do[AppendTo[arr2[z], NextPrime[3*Last[arr2[z]]]], {seqL}], {z, 2, 12}]; m = Map[arr2, Range[12]]; m // TableForm
t = Table[m[[n - k + 1]][[k]], {n, 12}, {k, n, 1, -1}] // Flatten (* Peter J. C. Moses, Sep 26 2013 *)

Incorrect comment deleted by Peter Munn, Aug 15 2017

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A055496 a(1) = 2; a(n) is smallest prime > 2*a(n-1).

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A194598 Union of A080359 and A164294.

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A229607 Square array read by antidiagonals downwards in which each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 2*p.

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A229609 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the greatest prime < 3*p.

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A229610 Array: each row starts with the least prime not in a previous row, and each prime p in a row is followed by the least prime > 3*p.

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