A005250 -id:A005250 - cp's OEIS frontend

A349995 Record gaps between odd squarefree semiprimes (A046388).

6, 12, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74, 84, 90, 92, 100, 102, 116, 118, 120, 132, 136, 138, 140, 142, 146, 152, 154, 156, 164, 170, 184, 186, 210

Offset: 1

Hugo Pfoertner, Dec 25 2021

			  n  A350098(n)  A350099(n)  a(n)
  1      15          21        6
  2      21          33       12
  3      95         111       16
  4     267         287       20
  5    2369        2391       22

Lucas A. Brown, Python program.

Records in A341828.
Cf. A350098 lower ends of the record gaps, A350099 upper ends of the record gaps.
Cf. A000101, A002386, A005250, A085809, A114057.

a(35)-a(41) from Lucas A. Brown, Feb 29 2024

A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.

Original entry on oeis.org

2, 3, 5, 11, 23, 30, 41, 50, 76, 100, 149, 159, 189, 345, 437, 509, 693, 1110, 1165, 5018, 14908, 18906, 19079, 28634, 38682, 80444, 105686, 185179, 265236, 269697, 409049, 558269, 1673629, 2965232, 3528015, 4292936, 34919969, 43957056, 148793437, 187220890, 424171123

Offset: 1

Hugo Pfoertner, Dec 25 2021

nonn

a(51) (in b-file) > 1.5*10^11, corresponding to A378904(51) > 723. - Hugo Pfoertner, Jan 04 2025

			  n  a(n)  p1   a(n)^2   p2   gap=2*A378904(n)
  1   2     3      4      5    2
  2   3     7      9     11    4
  3   5    23     25     29    6
  4  11   113    121    127   14
  5  23   523    529    541   18
  6  30   887    900    907   20
  7  41  1669   1681   1693   24
  8  50  2477   2500   2503   26

Hugo Pfoertner, Table of n, a(n) for n = 1..50

A378904 are the corresponding gaps, divided by 2.

Cf. A001223, A005250, A058043.

Module[{nn=4242*10^5,pg},pg=Table[{n,NextPrime[n^2]-NextPrime[n^2,-1]},{n,2,nn}];DeleteDuplicates[pg,GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* Harvey P. Dale, Jan 28 2023 *)

a350100(limit) = {my(pmax=0); for(k=2,limit, my(kk=k*k, pp=precprime(kk), pn=nextprime(kk), d=pn-pp); if(d>pmax, print1(k,", "); pmax=d))};
a350100(3000000)

from itertools import count, islice
from sympy import prevprime, nextprime
def A350100_gen(): # generator of terms
    c = 0
    for k in count(2):
        a = nextprime(m:=k**2)-prevprime(m)
        if a>c:
            yield k
            c = a
A350100_list = list(islice(A350100_gen(),20)) # Chai Wah Wu, Dec 17 2024

A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

Original entry on oeis.org

4, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132

Offset: 1

Enoch Haga, Mar 05 2000

nonn

Prime gaps associated with A053302.

a(17) is probably 1220 and a(19) is probably 1296. - Robert G. Wilson v, Mar 16 2004

			a(1) = 4 from 7 to 11. a(2) = 8 from 89 to 97. a(3) = 20 from 887 to 907.
a(5)=72 because the 5-digit prime 31397 begins a gap of 72.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Eric Weisstein's World of Mathematics, Prime Gaps

p_k's are in A053302. Cf. A005250, A002386. Essentially the same as A038460.

a(16) from Eric W. Weisstein, Mar 05 2004

A087103 Smallest jumping champion for prime(n).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 2, 2, 2, 2, 2, 6, 6, 6, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6

Offset: 2

Reinhard Zumkeller, Aug 10 2003

nonn

A number is called a jumping champion for n, if it is the most frequently occurring difference between consecutive primes <= n;

there are occasionally several jumping champions: see A087102; A087104(n) is the greatest jumping champion for prime(n).

T. D. Noe, Table of n, a(n) for n = 2..1001
A. Odlyzko, M. Rubinstein and M. Wolf, Jumping Champions
A. Odlyzko, M. Rubinstein and M. Wolf, Jumping Champions, Experimental Math., 8 (no. 2) (1999).
Eric Weisstein's World of Mathematics, Jumping Champion

Cf. A001223, A005250.

d=Table[0, {100}]; p=2; Table[q=NextPrime[p]; d[[q-p]]++; p=q; Position[d, Max[d]][[1,1]], {1000}]

A087104 Greatest jumping champion for prime(n).

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 2, 4, 2, 4, 4, 4, 4, 4, 4, 2, 2, 2, 4, 4, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6

Offset: 2

Reinhard Zumkeller, Aug 10 2003

nonn

A number is called a jumping champion for n, if it is the most frequently occurring difference between consecutive primes <= n;
there are occasionally several jumping champions: see A087102; A087103(n) is the smallest jumping champion for prime(n);
a(n)<=6 for small n, see Odlyzko et al. for primes>1.7*10^35.

T. D. Noe, Table of n, a(n) for n = 2..1001
A. Odlyzko, M. Rubinstein and M. Wolf, Jumping Champions
A. Odlyzko, M. Rubinstein and M. Wolf, Jumping Champions, Experimental Math., 8 (no. 2) (1999).
Eric Weisstein's World of Mathematics, Jumping Champion

Cf. A001223, A005250.

d=Table[0, {100}]; p=2; Table[q=NextPrime[p]; d[[q-p]]++; p=q; Position[d, Max[d]][[-1,1]], {1000}]

A100964 Smallest prime number that begins a prime gap of at least 2n.

Original entry on oeis.org

3, 7, 23, 89, 113, 113, 113, 523, 523, 887, 1129, 1327, 1327, 1327, 1327, 1327, 1327, 9551, 15683, 15683, 15683, 15683, 19609, 19609, 19609, 19609, 31397, 31397, 31397, 31397, 31397, 31397, 31397, 31397, 31397, 31397, 155921, 155921, 155921, 155921

Offset: 1

T. D. Noe, Nov 23 2004

nonn

			a(5) = a(6) = a(7) = 113 because there is a gap of 14 between 113 and 127.

Amiram Eldar, Table of n, a(n) for n = 1..775

Cf. A000230, A002386, A005250, A144309.

k=1; Table[While[Prime[k+1]-Prime[k] < 2n, k++ ]; Prime[k], {n, 48}]

lista(pmax) = {my(k = 1, prv = 2, m = 2, kprv = 2); forprime(p = 3, pmax, k++; if(p - prv >= m, for(i = 1, (p - prv - m)/2 + 1, print1(prv, ", ")); m = p - prv + 2; kprv = k); prv = p);} \\ Amiram Eldar, Sep 06 2024

a(n) = prime(A144309(n)). - Michel Marcus, Nov 02 2013

A104138 Smallest prime followed by n or more composites.

Original entry on oeis.org

2, 3, 7, 7, 23, 23, 89, 89, 113, 113, 113, 113, 113, 113, 523, 523, 523, 523, 887, 887, 1129, 1129, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 1327, 9551, 9551, 15683, 15683, 15683, 15683, 15683, 15683, 15683, 15683, 19609

Offset: 0

Lekraj Beedassy, Mar 07 2005

nonn

Except for a(1), records occur at even values of n, and each term appears an even number of times consecutively. (Proof. A maximal run of composites must begin and end at even numbers.) - Jonathan Sondow, May 31 2014

			a(10)=113 because it is the first prime occurring before primes 199,211,293,317,467,509,... all followed by at least ten successive composites.

Charles R Greathouse IV, Table of n, a(n) for n = 0..1000

Cf. A002386, A005250, A030296, A053695.

Record prime A002386(n+1) appears A053695(n-1) times, for n>1.

a(n) = A030296(n) - 1, for n > 0. - Jonathan Sondow, May 31 2014

a(34) corrected by Charles R Greathouse IV, Aug 09 2011

A107578 Prime index of A000101(n), maximal gap upper end prime index.

Original entry on oeis.org

2, 3, 5, 10, 25, 31, 100, 155, 190, 218, 1184, 1832, 2226, 3386, 14358, 30803, 31546, 40934, 103521, 104072, 149690, 325853, 1094422, 1319946, 2850175, 6957877, 10539433, 10655463, 20684333, 23163299, 64955635, 72507381

Offset: 1

Alex Beveridge, Apr 25 2007

nonn

Conjecture: log a(n) ~ n/2. That is, record prime gaps occur about twice as often as records in an i.i.d. random sequence of comparable length (see arXiv:1709.05508 for a heuristic explanation). - Alexei Kourbatov, Jan 18 2019

			The prime index of a(3) = 5, so prime(a(3)) = prime(5) = 11.

John W. Nicholson, Table of n, a(n) for n = 1..80 (terms 1..75 from Jens Kruse Andersen; further terms coming from Thomas R. Nicely site).
Alex Beveridge, Table giving known values of A000101(n), A005250(n), A107578(n)
Alexei Kourbatov, On the nth record gap between primes in an arithmetic progression, arXiv:1709.05508 [math.NT], 2017; Int. Math. Forum, 13 (2018), 65-78.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv:1901.03785 [math.NT], 2019.
Thomas R. Nicely, First occurrence prime gaps
Thomas R. Nicely, First occurrence prime gaps [Local copy, pdf only]
Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [See local copy in A007053]
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.

Cf. A000101, A005250, A002386, A005669.

a(n) = A005669(n)+1. - Jens Kruse Andersen, Oct 19 2010
From John W. Nicholson, Oct 29 2021: (Start)
a(n) = A000720(A000101(n)).
a(n) = A000720(A002386(n)) + 1. (End)

Name modified by John W. Nicholson, Nov 19 2013

A182877 Record gaps between Ramanujan primes.

Original entry on oeis.org

9, 12, 26, 46, 66, 86, 90, 100, 114, 120, 126, 174, 190, 212, 280, 386, 396, 410, 480, 598, 726, 916, 964, 1000, 1074, 1090, 1218, 1280, 1378, 1380, 1614, 1728, 1774, 1924, 1998, 2378, 2452, 2514, 2608, 2852, 2866, 2870, 3080

Offset: 1

T. D. Noe, Dec 09 2010

nonn

The Ramanujan prime beginning the gap is in A182876.

Table of n, a(n) for n = 1..43

Cf. A005250 (increasing gaps between primes).

a(37)-a(43) from Dana Jacobsen, Apr 29 2015

A239673 Record values in A239656 (the first differences of sphenic numbers).

Original entry on oeis.org

12, 24, 27, 28, 33, 35, 43, 44, 46, 48, 50, 52, 60, 65, 70, 72, 79, 82, 92, 98

Offset: 1

Reinhard Zumkeller, Mar 23 2014

Cf. A007304, A005250, A215231, A239656, A239674.

a239673 n = a239673_list !! (n-1)
(a239673_list, a239674_list) = unzip $ (12, 1) : f 1 12 a239656_list where
   f i v (q:qs) | q > v = (q, i) : f (i + 1) q qs
                | otherwise = f (i + 1) v qs
-- Reinhard Zumkeller, Mar 23 2014

lista(kmax) = {my(k1 = 30, d, dm = 0); forcomposite(k2 = k1 + 1, kmax, if(factor(k2)[,2] == [1,1,1]~, d = k2 - k1; if(d > dm, dm = d; print1(d, ", ")); k1 = k2));} \\ Amiram Eldar, May 19 2024

a(n) = A239656(A239674(n)).

a(12)-a(20) from Amiram Eldar, May 19 2024

cp's OEIS Frontend

A349995 Record gaps between odd squarefree semiprimes (A046388).

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A350100 Numbers k such that the prime gap between the consecutive primes p1 < k^2 < p2 sets a new record.

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Mathematica

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Python

A053303 Length of maximal prime gap p_{k+1} - p_k with starting prime p_k < 10^n.

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A087103 Smallest jumping champion for prime(n).

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Mathematica

A087104 Greatest jumping champion for prime(n).

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Mathematica

A100964 Smallest prime number that begins a prime gap of at least 2n.

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Mathematica

PARI

Formula

A104138 Smallest prime followed by n or more composites.

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A107578 Prime index of A000101(n), maximal gap upper end prime index.

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A182877 Record gaps between Ramanujan primes.