A329160 -id:A329160 - cp's OEIS frontend

A329158 Let P1>=3, P2, P3 be consecutive primes, with P3-P2=2. a(n)=(P2+P3)/12 when P2-P1 sets a record.

1, 2, 5, 25, 87, 192, 500, 1158, 1668, 4217, 4713, 5955, 17127, 28905, 61838, 76967, 96147, 139725, 260342, 1061923, 1205080, 4663498, 8871842, 11732765, 32534740, 42313103, 77638122, 92523718, 282054523, 728833340, 2940948542, 3344803093, 11810906035

Offset: 1

Hugo Pfoertner, Nov 06 2019

nonn

6*a(n)-1, 6*a(n)+1 are twin primes such that the prime gap immediately preceding 6*a(n)-1 sets a record. The corresponding gap lengths are provided in A329159.

Cf. A002822, A084105, A329159, A329160, A329161, A329164, A329165.

p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1((p2+p3)/12,", ")));p1=p2;p2=p3)

A329159 2*a(n) are the lengths of record prime gaps immediately preceding twin primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 11, 14, 15, 17, 20, 24, 26, 30, 35, 39, 42, 45, 50, 66, 68, 74, 75, 89, 98, 104, 107, 117, 123, 144, 159, 165, 180, 183, 192, 195

Offset: 1

Hugo Pfoertner, Nov 06 2019

The corresponding locations are provided in A329158. The gap 2 -> 3 before the pair 3, 5 is excluded.

Cf. A329158, A329160, A329161, A329164, A329165.

p1=3;p2=5;r=0;forprime(p3=7,1e9,if(p3-p2==2,d=p2-p1;if(d>r,r=d;print1(d/2,", ")));p1=p2;p2=p3)

a(31)-a(34) from Jinyuan Wang, Mar 01 2020

a(35) from David Trimas, Jul 26 2023

A329161 2*a(n) are the lengths of record prime gaps immediately following twin primes.

Original entry on oeis.org

2, 3, 5, 6, 9, 12, 14, 15, 17, 21, 26, 27, 29, 30, 35, 41, 45, 50, 59, 60, 63, 71, 74, 75, 84, 92, 96, 98, 104, 110, 125, 129, 144, 155, 156, 165, 180, 201, 204, 207

Offset: 1

Hugo Pfoertner, Nov 08 2019

The corresponding locations are provided in A329160, starting at the initial pair 5, 7.

			a(1) = 2 because the prime gap after 5, 7 has length 11 - 7 = 2*a(1) = 4,
a(2) = 3 because the prime gap after 29,31 has length 37 - 31 = 2*a(2) = 6, with location given by 6*A329160(2) = 30 +- 1. The gaps before all have length 4 (11,13 -> 17), (17,19 -> 23).

Cf. A329158, A329159, A329160, A329164, A329165.

p1=5; p2=7; r=0; forprime(p3=11, 1e9, if(p2-p1==2, d=p3-p2; if(d>r, r=d; print1(d/2, ", "))); p1=p2; p2=p3)

a(39)-a(40) from Jinyuan Wang, Mar 01 2020

A329164 Let P1, P2, P3, P4 be consecutive primes, with P2-P1=P4-P3=2. a(n)=(P1+P2)/12 when P3-P2 sets a new record.

Original entry on oeis.org

1, 23, 322, 495, 3407, 8113, 28893, 139708, 716182, 2497092, 5130198, 5761777, 7315173, 13194622, 145995245, 201544467, 417649822, 566513637, 833667068, 2266818768, 4710228962, 5186737183, 5192311957, 7454170028, 9853412390, 11817808908

Offset: 1

Hugo Pfoertner, Nov 07 2019

nonn

Position of record gaps with no primes bounded by consecutive pairs of twin primes. The length of the corresponding record gaps (P3-P1)/6 is given by A329165.

In the neighborhood of a(15), the growth of this sequence seems to change notably. See the plot2 graph in the links. Does this signify anything important? - Peter Munn, Aug 01 2025

			Values of P1, P2, P3, P4 corresponding to record gaps:
  P3-P1 P1   P2   P3   P4                   a(n)
   6     5    7   11   13        (5+7)/12 =   1
  12   137  139  149  151    (137+139)/12 =  23
  18  1931 1933 1949 1951  (1931+1933)/12 = 322
  30  2969 2971 2999 3001  (2969+2971)/12 = 495

Tomáš Brada and Natalia Makarova, Table of n, a(n) for n = 1..58
Peter Munn, Plot2 graph of sequence against Fibonacci growth

Cf. A053778, A329158, A329159, A329160, A329161, A329165.

p1=3;p2=5;p3=7;r=0;forprime(p4=11,1e9,if(p2-p1==2&&p4-p3==2,d=p3-p2;if(d>r,r=d;print1((p1+p2)/12,", ")));p1=p2;p2=p3;p3=p4)

A329165 Let P1, P2, P3, P4 be consecutive primes with P2-P1=P4-P3=2. a(n)=(P3-P1)/6 when the length of the gap with no primes between the two pairs of twin primes sets a record.

Original entry on oeis.org

1, 2, 3, 5, 6, 9, 12, 17, 18, 21, 22, 23, 25, 31, 33, 35, 40, 41, 42, 47, 48, 49, 51, 52, 53, 57, 58, 62, 63, 66, 71, 75, 77, 78, 81, 83, 85, 90, 91, 93, 98, 100, 105, 108, 111, 115, 119, 123, 125, 135, 138, 148, 150, 152, 165, 170, 173, 180

Offset: 1

Hugo Pfoertner, Nov 07 2019

The position of the occurrence of the n-th record is given by A329164(n)=(P1+P2)/12.

			See A329164.

Cf. A053778, A329158, A329159, A329160, A329161, A329164.

With[{s = Partition[Prime@ Range[10^5], 4, 1]}, Union@ FoldList[Max, Map[(#3 - #1)/6 & @@ # &, Select[s, #2 - #1 == #4 - #3 == 2 & @@ # &]]]] (* Michael De Vlieger, May 26 2020 *)

p1=3;p2=5;p3=7;r=0;forprime(p4=11,1e9,if(p2-p1==2&&p4-p3==2,d=p3-p1;if(d>r,r=d;print1(d/6,", ")));p1=p2;p2=p3;p3=p4)

a(27)-a(28) from Jinyuan Wang, Mar 01 2020

a(29)-a(58) found by Tomáš Brada, Natalia Makarova, May 12 2020

A084105 Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

Original entry on oeis.org

3, 29, 113, 139, 199, 523, 1151, 1669, 2971, 6947, 10007, 16141, 25471, 40639, 79699, 102761, 173359, 265621, 404851, 838249, 1349533, 1562051, 6371537, 7230479, 27980987, 42082303, 53231051, 70396589, 192983851, 253878617, 390932389, 465828731, 516540163, 1692327137

Offset: 1

Hugo Pfoertner, May 29 2003

nonn

Are there entries other than a(3) for which the smaller difference exceeds 2?

			a(3) = 113 because the ratio (113-109)/(127-113) = 2/7 = 0.28571.. is smaller than the previous minimum produced by (31-29)/(29-23) = 1/3 = 0.33333...

Martin Ehrenstein, Table of n, a(n) for n = 1..57
Hugo Pfoertner, Maximally asymmetric prime triples, FORTRAN program

Cf. A084106, A031132, A008996, A002386, A329158, A329159, A329160, A329161, A337489.

a084105(limit)={my(p1=2,p2=3,r=0);forprime(p3=5,limit,my(q=max((p2-p1)/(p3-p2),(p3-p2)/(p2-p1)));if(q>r,r=q;print1(p2,", "));p1=p2;p2=p3)};
a084105(600000000) \\ Hugo Pfoertner, Sep 04 2020

More terms from Don Reble, May 29 2003

a(32)-a(34) from Hugo Pfoertner, Nov 06 2019

A329252 Let P1 >= 5, P2, P3 be consecutive primes, with P2 - P1 = 2. a(n) = (P1 + P2)/12 for the first occurrence of (P3 - P2)/2 = n.

Original entry on oeis.org

1, 5, 0, 23, 33, 0, 322, 87, 0, 325, 278, 0, 495, 1293, 0, 2027, 4725, 0, 3468, 2690, 0, 27177, 14438, 0, 4245, 6773, 0, 13283, 24938, 0, 104283, 92067, 0, 28893, 60015, 0, 119362, 46905, 0, 44270, 106323, 0, 90713, 67475, 0, 266618, 207107, 0

Offset: 2

Hugo Pfoertner, Nov 10 2019

nonn

Position of first occurrence of a gap of length P3 - P2 = 2*n containing no primes, immediately following the twin primes (P1,P2). To indicate impossible gaps of lengths 8, 14, 20, ..., a(3k+1) is set to 0 for all k >= 1.

			a(5) = 23 because the prime gap following P1 = 6*23 - 1 = 137, P2 = 6*23 + 1 = 139 is the first such gap with length 2*n = 10. P3 - P2 = 149 - 139 = 10.

Hugo Pfoertner, Table of n, a(n) for n = 2..209

Cf. A329160, A329161, A329250, A329251.

my(v=vector(60), p1=5, p2=7, d); forprime(p3=11, 5e6, if(p2-p1==2, d=(p3-p2)/2; if(v[d]==0, v[d]=(p1+p2)/12)); p1=p2; p2=p3); v[2..49]

A337436 6a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2A007494(n).

Original entry on oeis.org

1, 5, 23, 33, 322, 87, 325, 278, 495, 1293, 2027, 4725, 3468, 2690, 27177, 14438, 4245, 6773, 13283, 24938, 104283, 92067, 28893, 60015, 119362, 46905, 44270, 106323, 90713, 67475, 266618, 207107, 139708, 1496910, 716182, 598867, 439633, 688518, 224922, 315893

Offset: 1

Hugo Pfoertner, Sep 02 2020

nonn

Apart from the atypical case [3, 5, 7], prime gaps nextprime(p+1)-p following a pair of twin primes p-2, p can only have the sizes 4, 6, 10, 12, 16, 18, ..., i.e., numbers k of the form 2*(k == 0 or 2 mod 3) = 2*A007494(n).

			a(1) = 1: The first occurrence of 3 consecutive primes [p-2, p, p+4] is at p = 6*a(1) + 1 = 7 -> [5, 7, 11],
a(2) = 5: consecutive primes [p-2, p, p+6] first occur at p = 6*a(2) * 1 = 31 -> [29, 31, 37],
a(3) = 23: consecutive primes [p-2, p, p+10] first occur at p = 6*a(3) + 1 = 139 -> [137, 139, 149].

Cf. A007497, A329160, A329161, A337435.

cp's OEIS Frontend

A329158 Let P1>=3, P2, P3 be consecutive primes, with P3-P2=2. a(n)=(P2+P3)/12 when P2-P1 sets a record.

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A329159 2*a(n) are the lengths of record prime gaps immediately preceding twin primes.

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A329161 2*a(n) are the lengths of record prime gaps immediately following twin primes.

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A329164 Let P1, P2, P3, P4 be consecutive primes, with P2-P1=P4-P3=2. a(n)=(P1+P2)/12 when P3-P2 sets a new record.

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A329165 Let P1, P2, P3, P4 be consecutive primes with P2-P1=P4-P3=2. a(n)=(P3-P1)/6 when the length of the gap with no primes between the two pairs of twin primes sets a record.

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A084105 Middle q of three consecutive primes p,q,r, such that one adjacent prime is near, the other is far and the ratio of the differences (whichever of (r-q)/(q-p) or (q-p)/(r-q) is greater than 1) sets a record.

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A329252 Let P1 >= 5, P2, P3 be consecutive primes, with P2 - P1 = 2. a(n) = (P1 + P2)/12 for the first occurrence of (P3 - P2)/2 = n.

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A337436 6*a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2*A007494(n).

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A337436 6a(n) + 1 is the least upper prime p of a pair of twin primes p - 2, p, for which the prime gap immediately following p achieves the size 2A007494(n).