A111871 -id:A111871 - cp's OEIS frontend

A111870 Prime p with prime gap q - p of n-th record merit, where q is smallest prime larger than p and the merit of a prime gap is (q-p)/log(p).

2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 2010733, 17051707, 20831323, 191912783, 436273009, 2300942549, 3842610773, 4302407359, 10726904659, 25056082087, 304599508537, 461690510011, 1346294310749, 1408695493609, 1968188556461, 2614941710599, 13829048559701, 19581334192423, 218209405436543, 1693182318746371

Offset: 1

N. J. A. Sloane, based on correspondence with Ed Pegg Jr, Nov 23 2005

nonn

As I understand it, the sequence refers to "Smallest prime p whose following gap has bigger merit than the other primes smaller than p." If that is the case, then it has an error. The sequence starts: 2, 3, 7, 113, 1129, 1327, 19609, 31397, 155921, 360653, 370261, 1357201, 4652353, 2010733, ... but you can see that 4652353 > 2010733, so in any case it should be listed after, not before it. But above that, its merit is 10.03 < 10.20, the merit of 2010733, so it is not in a mistaken position: it shouldn't appear in the sequence. - Jose Brox, Dec 31 2005
The logarithmic (base 10) graph seems to be linearly asymptotic to n with slope ~ 1/log(10) which would imply that: log(prime with n-th record merit) ~ n as n goes to infinity. - N. J. A. Sloane, Aug 27 2010
The sequence b(n) = (prime(n+1)/prime(n))^n is increasing for terms prime(n) of this sequence. - Thomas Ordowski, May 04 2012
The smallest prime(n) such that (prime(n+1)/prime(n))^n is increasing: 2, 3, 7, 23, 113, 1129, 1327, ... (A205827). - Thomas Ordowski, May 04 2012
(prime(n+1)/prime(n))^n > 1 + merit(n) for n > 2, where merit(n) = (prime(n+1)-prime(n))/log(prime(n)). - Thomas Ordowski, May 14 2012
Merit(1) + merit(2) + ... + merit(n) =: S(n) ~ n, where merit(n) is as above. - Thomas Ordowski, Aug 03 2012
For the index of a(n), see the comment at A214935. - John W. Nicholson, Nov 21 2013

			The first few entries correspond to the following gaps. The table gives n, p, gap = q-p and the merit of the gap.
   1,       2,   1, 1.4427
   2,       3,   2, 1.82048
   3,       7,   4, 2.05559
   4,     113,  14, 2.96147
   5,    1129,  22, 3.12985
   6,    1327,  34, 4.72835
   7,   19609,  52, 5.26116
   8,   31397,  72, 6.95352
   9,  155921,  86, 7.19238
  10,  360653,  96, 7.50254
  11,  370261, 112, 8.73501
  12, 1357201, 132, 9.34782

Ed Pegg, Jr., Posting to Seq Fan mailing list, Nov 23 2005

Hugo Pfoertner, Table of n, a(n) for n = 1..39, extracted from pzktupel link.
Jens Kruse Andersen, The Top-20 Prime Gaps
Jens Kruse Andersen, Maximal gaps
Jens Kruse Andersen, Record prime gaps, current version maintained by Norman Luhn.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

For the gaps, see A111871.

Cf. A002386, A111943, A214935, A277552.

With[{s = Map[(#2 - #1)/Log[#1] & @@ # &, Partition[Prime@ Range[10^6], 2, 1]]}, Map[Prime@ FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Jul 19 2018 *)

a(n) = A277552(n) - A111871(n). - Bobby Jacobs, Nov 13 2016

Corrected by Jose Brox, Dec 31 2005
Corrected and edited by Daniel Forgues, Oct 23 2009
Further edited by Daniel Forgues, Nov 01 2009, Nov 13 2009, Nov 24 2009

A114403 Triprime gaps. First differences of A014612.

Original entry on oeis.org

4, 6, 2, 7, 1, 2, 12, 2, 1, 5, 2, 11, 3, 2, 2, 5, 1, 2, 14, 6, 1, 3, 3, 5, 4, 2, 1, 7, 1, 5, 8, 9, 1, 5, 1, 10, 1, 5, 1, 1, 2, 1, 7, 4, 2, 2, 5, 12, 5, 10, 8, 1, 5, 2, 4, 2, 1, 1, 9, 3, 3, 5, 2, 5, 2, 4, 3, 2, 1, 1, 4, 2, 18, 6, 2, 4, 3, 7, 1, 5, 5, 2, 9, 2, 1

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 4 = 12-8 where 8 is the first triprime and 12 is the second.
a(2) = 6 = 18-12
a(3) = 2 = 20-18
a(4) = 7 = 27-20

T. D. Noe, Table of n, a(n) for n=1..10000

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

is3Alm := proc(n::integer) local ifa,ex,i ; ifa := op(2,ifactors(n)) ; ex := 0 ; for i from 1 to nops(ifa) do ex := ex+ op(2,op(i,ifa)) ; od : if ex = 3 then RETURN(true) ; else RETURN(false) ; fi ; end: A014612 := proc(n::integer) local resul,i; i :=1; resul := 8 ; while i < n do resul := resul + 1 ; if is3Alm(resul) then i := i+1 ; fi ; od ; RETURN(resul) ; end: A114403 := proc(n::integer) RETURN(A014612(n+1)-A014612(n)) ; end: for n from 1 to 160 do printf("%d,",A114403(n)) ; od: # R. J. Mathar, Apr 25 2006

Differences[Select[Range[425], PrimeOmega[#] == 3 &]] (* Jayanta Basu, Jul 01 2013 *)

a(n) = A014612(n+1) - A014612(n).

Corrected and extended by R. J. Mathar, Apr 25 2006

A114412 Records in semiprime gaps ordered by merit.

Original entry on oeis.org

2, 3, 4, 6, 11, 19, 24, 28, 30, 32, 38, 47, 54, 70, 74, 107, 110, 112, 120, 126, 146

Offset: 1

Jonathan Vos Post, Nov 25 2005

There is an associated index list n = 1, 2, 4, 6, 34, 422, 1765, 4585, 8112, 8650, 8861, 75150, ... and an associated semiprime list A001358(n) = 4, 6, 10, 15, 1418, 6559, 17965, 32777, 35103, 35981, 340894, ... - R. J. Mathar, Mar 15 2009

			Records defined in terms of A065516 and A001358:
.
  n  A065516(n)  A065516(n)/log_10(A001358(n))
  =  ==========  ==============================
  1       2      2 / log_10(4)  = 3.32192809...
  2       3      3 / log_10(6)  = 3.85529162...
  3       1      1 / log_10(9)  = 1.04795163...
  4       4      4 / log_10(10) = 4.00000000
  5       1      1 / log_10(14) = 0.87250286...
  6       6      6 / log_10(15) = 5.10164492...
  7       1      1 / log_10(21) = 0.75630419...
  8       3      3 / log_10(22) = 2.23476557...
  9       1      1 / log_10(25) = 0.71533827...

Eric Weisstein's World of Mathematics, Semiprime.
Eric Weisstein's World of Mathematics, Prime Gaps.

Cf. A001358, A065516, A111870, A111871, A113688-A113693, A114403-A114411, A114412-A114422.

sp = 4; m0 = 0;  l = {}; lim = 1000000;
For[i = 5, i <= lim, i++, If[PrimeOmega[i] == 2, m = (i - sp)/Log[sp]; If[m > m0, m0 = m; AppendTo[l, i - sp]]; sp = i] ]; l (* Robert Price, Oct 29 2018 *)

a(n) = records in A065516(n)/log_10(A001358(n)) = records in (A001358(n+1) - A001358(n))/log_10(A001358(n)).

Corrected and extended by Charles R Greathouse IV, Oct 05 2006

a(16)-a(21) from Donovan Johnson, Feb 17 2010

A114405 5-almost prime gaps. First differences of A014614.

Original entry on oeis.org

16, 24, 8, 28, 4, 8, 42, 6, 8, 4, 20, 8, 35, 9, 12, 6, 2, 8, 20, 4, 8, 56, 10, 14, 4, 9, 3, 12, 20, 10, 6, 8, 4, 28, 4, 20, 32, 15, 21, 4, 2, 18, 4, 14, 26, 4, 15, 5, 4, 4, 8, 4, 2, 26, 16, 6, 2, 8, 20, 48, 20, 34, 6, 3, 27, 2, 4, 20, 1, 7, 16, 8, 4, 4, 6, 30, 6, 6, 12, 6, 3, 11

Offset: 1

Jonathan Vos Post, Nov 25 2005

First occurrences of a(n)=1,2,3,.. are at n=69, 17, 27, 5, 48, 8, 70, 3, 14, 23, 82, 15, 150, 24, 38, 1, 172, 42, 258, 11, 39, 135, 102, 2, 779, 45, 65, 4, 518, 76, 263, 37, 211, 62, 13, 1009, 2463, 606, 254, 151, 3348, 7, 4513,... - R. J. Mathar, Oct 06 2007

			a(1) = 16 = 48-32 where 32 is the first 5-almost prime and 48 is the second.
a(2) = 24 = 72-48.
a(3) = 8 = 80-72.
a(4) = 28 = 108-80.
a(5) = 4 = 112-108.
a(6) = 8 = 120-112.
a(7) = 42 = 162-120.
a(8) = 6 = 168-162.
a(13) = 35 = 243-208.
a(22) = 56 = 368-312.

R. J. Mathar, Table of n, a(n) for n = 1..9999

Cf. A014614, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

Differences[Select[Range[2000],PrimeOmega[#]==5&]] (* Harvey P. Dale, Sep 28 2019 *)

a(n) = A014614(n+1) - A014614(n).

More terms from R. J. Mathar, Oct 06 2007

A114414 Records in 4-almost prime gaps ordered by merit.

Original entry on oeis.org

8, 12, 14, 21, 28

Offset: 1

Jonathan Vos Post, Nov 25 2005

Next term (if it exists) associated with A014613 > 1030000. - R. J. Mathar, Mar 13 2007

			Records defined in terms of A114404 and A014613:
  n  A114404(n)  A114404(n)/log_10(A014613(n))
  =  ==========  =============================
  1      8       8/log_10(16)   = 6.64385619
  2      12      12/log_10(24)  = 8.6943213
  3      4       4/log_10(36)   = 2.57019442
  4      14      14/log_10(40)  = 8.73874891
  5      2       2/log_10(54)   = 1.15447195
  6      4       4/log_10(56)   = 2.2880834
  7      21      21/log_10(60)  = 11.810019
  ...
  13     22      22/log_10(104) = 10.9071078
  ...
  21     28      28/log_10(156) = 12.7671725

Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

Digits := 16 : A114414 := proc() local n,a014613,a114414,rec ; a014613 := 16 ; a114414 := 8 ; rec := a114414/log(a014613) ; print(a114414) ; n := 17 ; while true do while numtheory[bigomega](n) <> 4 do n := n+1 ; od ; a114414 := n-a014613 ; if ( evalf(a114414/log(a014613)) > evalf(rec) ) then rec := a114414/log(a014613) ; print(a114414) ; fi ; a014613 := n ; n := n+1 : od ; end: A114414() ; # R. J. Mathar, Mar 13 2007

a(n) = records in A114404(n)/log_10(A014613(n)) = records in (A014613(n+1) - A014613(n))/log_10(A014613(n)).

A114404 4-almost prime gaps. First differences of A014613.

Original entry on oeis.org

8, 12, 4, 14, 2, 4, 21, 3, 4, 2, 10, 4, 22, 6, 3, 1, 4, 10, 2, 4, 28, 5, 7, 2, 6, 6, 10, 5, 3, 4, 2, 14, 2, 10, 16, 18, 2, 1, 9, 2, 7, 13, 2, 10, 2, 2, 4, 2, 1, 13, 8, 3, 1, 4, 10, 24, 10, 17, 3, 15, 1, 2, 10, 4, 8, 4, 2, 2, 3, 15, 3, 3, 6, 3, 7, 4, 10, 4, 8, 6, 4, 2, 2, 8, 4, 1, 35, 1, 4, 7, 4, 8, 6

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 8 = 24-16 where 16 is the first 4-almost prime and 24 is the second.
a(2) = 12 = 36-24.
a(3) = 4 = 40-36.
a(4) = 14 = 54-40.
a(5) = 2 = 56-54.
a(6) = 4 = 60-56.
a(7) = 21 = 81-60.
a(13) = 22 = 126-104.
a(21) = 28 = 184-156.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A014613, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A114404 := proc(nmax) local a,i,a014613 ; a := [] ; i := 1 ; a014613 := -1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 4 then if a014613 > 0 then a := [op(a),i-a014613] ; fi ; a014613 := i ; fi ; i := i+1 ; end: a ; end: A114404(200) ; # R. J. Mathar, May 10 2007

Differences[Select[Range[800],Total[FactorInteger[#][[All,2]]]==4&]] (* Harvey P. Dale, Feb 14 2017 *)
Select[Range[1000],PrimeOmega[#]==4&]//Differences (* Harvey P. Dale, May 12 2018 *)

a(n) = A014613(n+1) - A014613(n).

Corrected and extended by R. J. Mathar, May 10 2007

A114406 6-almost prime gaps. First differences of A046306.

Original entry on oeis.org

32, 48, 16, 56, 8, 16, 84, 12, 16, 8, 40, 16, 70, 18, 24, 12, 4, 16, 40, 8, 16, 105, 7, 20, 28, 8, 18, 6, 24, 40, 20, 12, 16, 8, 56, 8, 40, 64, 30, 42, 8, 4, 27, 9, 8, 28, 52, 8, 30, 10, 8, 8, 16, 8, 4, 52, 32, 12, 4, 16, 40, 96, 40, 5, 63, 12, 6, 54, 4, 8, 40, 2, 14, 32, 16, 8, 8, 12, 45

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 32 = 96-64 where 64 is the first 6-almost prime and 96 is the second.
a(2) = 48 = 144-96.
a(3) = 16 = 160-144.
a(4) = 56 = 216-160.
a(5) = 8 = 224-216.
a(6) = 16 = 240-224.
a(7) = 84 = 324-240.
a(8) = 12 = 336-324.
a(22) = 105 = 729-624.

Zak Seidov, Table of n, a(n) for n = 1..10000

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

a(n) = A046306(n+1) - A046306(n).

More terms from R. J. Mathar, Aug 31 2007

A114407 7-almost prime gaps. First differences of A046308.

Original entry on oeis.org

64, 96, 32, 112, 16, 32, 168, 24, 32, 16, 80, 32, 140, 36, 48, 24, 8, 32, 80, 16, 32, 210, 14, 40, 56, 16, 36, 12, 48, 80, 40, 24, 32, 16, 112, 16, 80, 107, 21, 60, 84, 16, 8, 54, 18, 16, 56, 104, 16, 60, 20, 16, 16, 32, 16, 8, 104, 64, 24, 8, 32, 80, 192, 80, 10, 126, 24, 12

Offset: 1

Jonathan Vos Post, Nov 25 2005

			a(1) = 64 = 192-128 where 128 is the first 7-almost prime and 192 is the second.
a(2) = 96 = 288-192.
a(3) = 32 = 320-288.
a(4) = 112 = 432-320.
a(5) = 16 = 448-432.
a(6) = 32 = 480-448.
a(7) = 168 = 648-480.
a(8) = 24 = 672-648.

Zak Seidov, Table of n, a(n) for n = 1..10000

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

Differences[Select[Range[10000],PrimeOmega[#]==7&]] (* Harvey P. Dale, Oct 13 2019 *)

a(n) = A046308(n+1) - A046308(n).

Corrected and extended by R. J. Mathar, Aug 31 2007

A114408 8-almost prime gaps. First differences of A046310.

Original entry on oeis.org

128, 192, 64, 224, 32, 64, 336, 48, 64, 32, 160, 64, 280, 72, 96, 48, 16, 64, 160, 32, 64, 420, 28, 80, 112, 32, 72, 24, 96, 160, 80, 48, 64, 32, 224, 32, 160, 214, 42, 120, 168, 32, 16, 108, 36, 32, 112, 208, 32, 120, 40, 32, 32, 64, 32, 16, 208, 128, 48

Offset: 1

Jonathan Vos Post, Dec 03 2005

			a(1) = 128 = 384-256 = A046310(2) - A046310(1).
a(2) = 192 = 576-384.
a(3) = 64 = 640-576.
a(4) = 224 = 864-640.
a(5) = 32 = 896-864.
a(6) 64 = 960-896.
a(7) = 336 = 1296-960.
a(8) = 48 = 1344-1296.
a(22) = 420 = 2916-2496.

Zak Seidov, Table of n, a(n) for n = 1..10000

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

a(n) = A046310(n+1) - A046310(n).

A114415 Records in 5-almost prime gaps ordered by merit.

Original entry on oeis.org

16, 24, 28, 42, 56, 70

Offset: 1

Jonathan Vos Post, Nov 25 2005

Next term, if it exists, is associated with indices above 100000 in A114405 and A014614. - R. J. Mathar, May 10 2007

			Records defined in terms of A114405 and A014614:
  n  A114405(n)  A114405(n)/log_10(A014614(n))
  =  ==========  =============================
  1      16      16/log_10(32)  = 10.6301699
  2      24      24/log_10(48)  = 14.2751673
  3      8       8/log_10(72)   = 4.30725248
  4      28      28/log_10(80)  = 14.7129144
  5      4       4/log_10(108)  = 1.96712564
  6      8       8/log_10(112)  = 3.90392819
  7      42      42/log_10(120) = 20.2002592
  8      6       6/log_10(168)  = 2.69625443
  ...
  22     56      56/log_10(312) = 22.4524976

Cf. A014614, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

A014614 := proc(nmax) local a,i; a := [] ; i := 1 ; while nops(a) < nmax do if numtheory[bigomega](i) = 5 then a := [op(a),i] ; fi ; i := i+1 ; end: a ; end: A114405 := proc(a014614) local a,i; a := [] ; for i from 2 to nops(a014614) do a := [op(a), op(i,a014614)-op(i-1,a014614)] ; od ; a ; end: a014614 := A014614(100000) : a114405 := A114405(a014614) : Digits := 30 : rec := -1 : for i from 1 to nops(a114405) do if evalf(a114405[i]/log(a014614[i])) > rec then printf("%d, ",a114405[i]) ; rec := evalf(a114405[i]/log(a014614[i])) ; fi ; od ; # R. J. Mathar, May 10 2007

a(n) = records in A114405(n)/log_10(A014614(n)) = records in (A014614(n+1) - A014614(n))/log_10(A014614(n)).

a(6) from R. J. Mathar, May 10 2007

cp's OEIS Frontend

A111870 Prime p with prime gap q - p of n-th record merit, where q is smallest prime larger than p and the merit of a prime gap is (q-p)/log(p).

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A114403 Triprime gaps. First differences of A014612.

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A114412 Records in semiprime gaps ordered by merit.

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A114405 5-almost prime gaps. First differences of A014614.

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A114414 Records in 4-almost prime gaps ordered by merit.

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A114404 4-almost prime gaps. First differences of A014613.

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A114406 6-almost prime gaps. First differences of A046306.

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