A111870 -id:A111870 - cp's OEIS frontend

A277552 Primes q with prime gap q - p of n-th record merit.

3, 5, 11, 127, 1151, 1361, 19661, 31469, 156007, 360749, 370373, 1357333, 2010881, 17051887, 20831533, 191913031, 436273291, 2300942869, 3842611109, 4302407713, 10726905041, 25056082543, 304599509051, 461690510543, 1346294311331, 1408695494197, 1968188557063, 2614941711251, 13829048560417, 19581334193189

Offset: 1

Bobby Jacobs, Nov 07 2016

nonn

The merit of the gap between consecutive primes p and q is (q-p)/log(p).

This sequence is infinite (Westzynthius 1931). - Charles R Greathouse IV, Nov 10 2016

Cf. A111870, A111871.

p = 2; q = 3; lmt = 0; lst = {}; While[p < 10^12, If[q > p + lmt*Log[p], AppendTo[lst, q]; Print[q]; lmt = (q - p)/Log[p]]; p = q; q = NextPrime@ p]; lst (* or *)
(* set lst = the terms in A111870 *) NextPrime[ lst] (* Robert G. Wilson v, Nov 07 2016 *)

r=rm=0; p=2; forprime(q=3,, t=q-p; if(t>r, r=t; t/=log(p); if(t>rm, rm=t; print1(q", "))); p=q) \\ Charles R Greathouse IV, Nov 11 2016

a(n) = A111870(n) + A111871(n). - Charles R Greathouse IV, Nov 11 2016

More terms from Robert G. Wilson v, Nov 07 2016

A114417 Records in 7-almost prime gaps, ordered by merit.

Original entry on oeis.org

64, 96, 112, 168, 210, 280

Offset: 1

Jonathan Vos Post, Nov 25 2005

			Records defined in terms of A114407 and A046308:
n A114407(n) A114407(n)/log(A046308(n))
1 64 64/log 128 = 30.371914
2 96 96/log 192 = 42.0443868
3 32 32/log 288 = 13.0113433
4 112 112/log 320 = 44.7079021
5 16 16/log 432 = 6.07099172
6 32 32/log 448 = 12.0696509
7 168 168/log 480 = 62.6575474
8 24 24/log 648 = 8.53614076

Cf. A046308, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = Records in A114417(n)/log(A046308(n)) = Records in (A046308(n+1) - A046308(n))/log(A046308(n)).

a(5)-a(6) from Donovan Johnson, Feb 17 2010

A114418 Records in 8-almost prime gaps ordered by merit.

Original entry on oeis.org

128, 192, 224, 336, 420, 560

Offset: 1

Jonathan Vos Post, Dec 03 2005

			Records defined in terms of A114408 and A046310:
n A114418(n) A114418(n)/log(A046310(n)).
1 128 128/log 256 = 53.1508495
2 192 192/log 384 = 74.2938824
3 64 64/log 576 = 23.1848568
4 224 224/log 640 = 79.8238182
5 32 32/log 864 = 10.8972758
6 64 64/log 896 = 21.6779549
7 336 336/log 960 = 112.665809
8 48 48/log 1296 = 15.4211665
22 420 420/log 2496 = 123.629603

Cf. A046310, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = records in A114418(n)/log(A046310(n)) = records in (A046310(n+1) - A046310(n))/log(A046310(n)).

Offset corrected and a(6) from Donovan Johnson, Feb 17 2010

A228775 a(n) is the maximal k>=1 such that nextprime(jn)<=(j+1)n, j=1,...,k.

Original entry on oeis.org

2, 3, 7, 5, 17, 14, 16, 24, 12, 19, 28, 43, 86, 80, 34, 82, 78, 73, 69, 66, 117, 329, 57, 222, 171, 228, 178, 470, 291, 359, 505, 366, 585, 576, 644, 544, 423, 742, 502, 636, 765, 466, 936, 578, 697, 682, 541, 1442, 640, 627, 615, 603, 2025, 1660, 570, 1833

Offset: 1

Vladimir Shevelev, Sep 04 2013

nonn

			If n=3, then, for j=1, nextprime(3)<=6; for j=2, nextprime(6)<=9; for j=3,nextprime(9)<=12; for j=4, nextprime(12)<=15; for j=5, nextprime(15)<=18; for j=6,nextprime(18)<=21; for j=7, nextprime(21)<=24, BUT for j=8, nextprime(24)>27. Thus a(3)=7.

Main sequence is A110835.

Cf. A002386, A005250, A111870.

a[n_] := For[k = 1, True, k++, If[NextPrime[k*n] <= (k+1)*n && NextPrime[(k+1)*n] > (k+2)*n, Return[k]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Sep 05 2013 *)

Conjectural inequality: for n>=2, a(n) <= log^2(n*a(n)). This essentially corresponds to Cramer's conjecture for prime gaps.

More terms from Peter J. C. Moses

A114413 Records in 3-almost prime gaps ordered by merit.

Original entry on oeis.org

4, 6, 12, 58, 83

Offset: 1

Jonathan Vos Post, Nov 25 2005

			Records defined in terms of A114403 and A014612:
  n  A114403(n)  A114403(n)/log_10(A014612(n))
  =  ==========  =============================
  1      4       4/log_10(8)   = 4.42923746
  2      6       6/log_10(12)  = 5.55977045
  3      2       2/log_10(18)  = 1.59327954
  4      7       7/log_10(20)  = 5.38035251
  5      1       1/log_10(27)  = 0.698634425
  6      2       2/log_10(28)  = 1.38201907
  7      12      12/log_10(30) = 8.12390991
  ...
  19     14      14/log_10(78) = 7.3992072

Cf. A014612, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = records in A114403(n)/log_10(A014612(n)) = records in (A014612(n+1) - A014612(n))/log_10(A014612(n)).

a(4)-a(5) from Donovan Johnson, Feb 17 2010

A114416 Records in 6-almost prime gaps ordered by merit.

Original entry on oeis.org

32, 48, 56, 84, 105, 140

Offset: 1

Jonathan Vos Post, Nov 25 2005

			Records defined in terms of A114406 and A046306:
n A114406(n) A114406(n)/log(A046306(n)).
1 32 32/log 64 = 17.7169498
2 48 48/log 96 = 24.2146479
3 16 16/log 144 = 7.41302726
4 56 56/log 160 = 25.4069653
5 8 8/log 216 = 3.42692589
6 16 16/log 224 = 6.80779215
7 84 84/log 240 = 35.2909853
8 12 12/log 324 = 4.77983862
...
22 105 105/log 624 = 37.5646032

Cf. A046306, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

a(n) = records in A114406(n)/log(A046306(n)) = records in (A046306(n+1) - A046306(n))/log(A046306(n)).

a(6) from Donovan Johnson, Feb 17 2010

A316917 Let g(n) be the n-th maximal prime gap; a(n) = 1 if g(n) has record merit, 0 if it does not.

Original entry on oeis.org

1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1

Offset: 1

Rodolfo Ruiz-Huidobro, Jul 16 2018

a(n) = 1 if A002386(n) is in A111870, a(n) = 0 if A002386(n) is not in A111870.
The merit M of a prime gap of measure g following the prime p_1 is defined as M=g/ln(p_1). It is the ratio of the measure of the gap to the "average" measure of gaps near that point. As an example, the merit of the sixth maximal gap, of size 14, after prime 113 is 2.96.
a(81) = 0 because there are previous maximal gaps with higher merits. - Rodolfo Ruiz-Huidobro, Jan 23 2024
a(82) = 1 as the merit of the gap is 1572/log(18571673432051830099)=1572/44.37=35.43 (which is a record merit). - Rodolfo Ruiz-Huidobro, May 10 2024
a(83) =1 as the merit for the gap is 1676/log(20733746510561444539) =1676/44.48=37.681 (which is a record merit). - Rodolfo Ruiz-Huidobro, Dec 20 2024

			The 5th record prime gap from 89 to 97 does not have record merit, so a(5) = 0.
The 10th record prime gap from 1327 to 1361 has record merit, so a(10) = 1.

Prime Gap List Community, Record prime gaps, 2021.
Index entries for characteristic functions

Cf. A000101, A002386, A111870, A111871, A005250.

Block[{nn = 10^6, s, t, u, v}, s = Prime@ Range[nn]; t = Differences@ s; u = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[nn], 2, 1]]; v = Map[Prime@ FirstPosition[u, #][[1]] &, Union@ FoldList[Max, u]]; Boole[! FreeQ[v, s[[FirstPosition[t, #][[1]] ]] ] ] & /@ Union@ FoldList[Max, t]] (* Michael De Vlieger, Jul 19 2018 *)

a(81) from Rodolfo Ruiz-Huidobro, Jan 23 2024
a(82) from Rodolfo Ruiz-Huidobro, May 10 2024
a(83) from Rodolfo Ruiz-Huidobro, Dec 09 2024

cp's OEIS Frontend

A277552 Primes q with prime gap q - p of n-th record merit.

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A114417 Records in 7-almost prime gaps, ordered by merit.

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A114418 Records in 8-almost prime gaps ordered by merit.

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A228775 a(n) is the maximal k>=1 such that nextprime(j*n)<=(j+1)*n, j=1,...,k.

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A114413 Records in 3-almost prime gaps ordered by merit.

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A114416 Records in 6-almost prime gaps ordered by merit.

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A316917 Let g(n) be the n-th maximal prime gap; a(n) = 1 if g(n) has record merit, 0 if it does not.

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A228775 a(n) is the maximal k>=1 such that nextprime(jn)<=(j+1)n, j=1,...,k.