A000230 -id:A000230 - cp's OEIS frontend

A140791 First occurrence of prime gap 10*n.

139, 887, 4297, 19333, 31907, 43331, 173359, 542603, 404851, 396733, 1468277, 1895359, 5518687, 7621259, 13626257, 33803689, 27915737, 17051707, 142414669, 378043979, 20831323, 47326693, 607010093, 391995431, 387096133, 944192807

Offset: 1

Lekraj Beedassy, Jul 14 2008

nonn

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 139, pp 47, Ellipses, Paris 2008.

Index entries for primes, gaps between.

Cf. A101232, A062529, A000230.

a(n)=A000230(5n), a(3n) = A224522(n) = A058193(5n). - M. F. Hasler, Apr 09 2013

Offset corrected by Arkadiusz Wesolowski, Jun 07 2011

A058193 Smallest prime p such that there is a gap of 6n between p and the next prime.

Original entry on oeis.org

23, 199, 523, 1669, 4297, 9551, 16141, 28229, 35617, 43331, 162143, 31397, 188029, 461717, 404851, 360653, 1444309, 2238823, 492113, 1895359, 1671781, 1357201, 3826019, 11981443, 13626257, 17983717, 39175217, 37305713, 52721113

Offset: 1

Labos Elemer, Nov 28 2000

nonn

			d = 72 appears after 31397, while smaller d = 54, 60, 66 come later, following primes 35617, 43331, 162143, respectively.

Amiram Eldar, Table of n, a(n) for n = 1..240
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Index entries for primes, gaps between.

Cf. A101232, A062529, A000230.

Module[{nn=32*10^5,prs,gps},prs=Prime[Range[nn]];gps=Differences[prs];Table[SelectFirst[Thread[{Most[prs],gps}],#[[2]]==6n&],{n,30}]][[;;,1]] (* Harvey P. Dale, Mar 03 2025 *)

a(n) = {p=3; q = nextprime(p+1); while((q-p) != 6*n, p = q; q = nextprime(q+1)); p;} \\ Michel Marcus, Mar 12 2016

a(n) = A000230(3n).

Offset corrected by M. F. Hasler, Apr 09 2013

A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

Original entry on oeis.org

2, 3, 7, 89, 1831, 5591, 89689, 3851459, 1872851947, 1999066711391, 22790428875364879

Offset: 0

Labos Elemer, Jun 25 2001

nonn

a(11) <= 79419801290172271035479303914142441 and  a(12) <= 55128448018333565337014555712123010955456071077000028555991469751. - Abhiram R Devesh, Aug 09 2014
From Zhining Yang, Dec 02 2022: (Start)
a(11) = 5333419265419188034369535864125349, 34 digits, discovered by Helmut Spielauer in 2013
a(12) = 55128448018333565337014555712123010955456071077000028555991469751, 65 digits, discovered by Helmut Spielauer in 2013
a(13) = 192180552346991956641101827551986346298837407139466361414211497406670710665021150917759713696699494356609164354068319457039591759, 129 digits, discovered by Dana Jacobsen in 2016
a(14) = 267552521*631#/210 - 9606, 268 digits, discovered by Dana Jacobsen in 2016
a(15) = 2717*1303#/268590 - 16670, 552 digits, discovered by Dana Jacobsen in 2014
a(16) = 7079*3559#/9870 - 36310, 1517 digits, discovered by Michiel Jansen, Pierre Cami, and Jens Kruse Andersen in 2013
a(17) = 1111111111111111111*9059#/(11#*5237) - 86522, 3899 digits, discovered by Hans Rosenthal in 2017
a(11) to a(17) were searched from Thomas R. Nicely's homepage. (End)
Importantly, the values in the previous comment are only upper bounds on a(11)-a(17), and are (almost certainly) not the correct values. As of this comment, the largest prime gap length whose first occurrence is known is 1676 < 2^11. - Brian Kehrig, May 01 2025

			a(2)=7 because 7 and 11 are consecutive primes with difference 2^2=4.
a(3)=89 because 89 and 97 are consecutive primes with difference 2^3=8.

Cino Hilliard, TwinPrimes Java code.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101].
Thomas R. Nicely, Other tables of prime gaps

Cf. A000230, A062530, A101232, A002386.

f[n_] := Block[{k = 1}, While[Prime[k + 1] != n + Prime[k], k++ ]; Prime[k]]; Do[ Print[ f[2^n]], {n, 0, 10}] (* Robert G. Wilson v, Jan 13 2005 *)

import sympy
n=0
while n>=0:
    p=2
    while sympy.nextprime(p)-p!=(2**n):
        p=sympy.nextprime(p)
    print(p)
    n=n+1
    p=sympy.nextprime(p)
## Abhiram R Devesh, Aug 09 2014

a(n) = A000230(2^(n-1)). - R. J. Mathar, Jan 12 2007

a(n) = A000230(2^(n-1)) = Min{p|nextprime(p)-p = 2^n} [may need adjusting since offset has been changed].

a(10) sent by Robert G. Wilson v, Jan 13 2005

a(11)-a(12) removed by Brian Kehrig, May 01 2025

A079016 Suppose p and q = p+12 are primes. Define the difference pattern of (p,q) to be the successive differences of the primes in the range p to q. There are 14 possible difference patterns, namely [12], [2,10], [4,8], [6,6], [8,4], [10,2], [2,4,6], [2,6,4], [4,2,6], [4,6,2], [6,2,4], [6,4,2], [2,4,2,4] and [4,2,4,2]. Sequence gives smallest value of p for each difference pattern, sorted by magnitude.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 47, 67, 89, 137, 139, 199, 397, 1601

Offset: 1

Labos Elemer, Jan 24 2003

			p=1601, q=1613 has difference pattern [6,2,4] and {1601,1607,1609,1613} is the corresponding consecutive prime 4-tuple.

A022006(1)=5, A022007(1)=7, A078847(1)=17, A078851(1)=19, A078848(1)=29, A078855(1)=31, A047948(1)=47, A078850(1)=67, A031930(1)=A000230(6)=199, A046137(1)=7, A078853(1)=1601.

Cf. A079017-A079024.

Function[s, Function[t, Union@ Flatten@ Map[s[[First@ Position[t, #]]] &, {{12}, {2, 10}, {4, 8}, {6, 6}, {8, 4}, {10, 2}, {2, 4, 6}, {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4}, {6, 4, 2}, {2, 4, 2, 4}, {4, 2, 4, 2}}]]@ Map[Differences@ Select[Range[#, # + 12], PrimeQ] &, s]]@ Select[Prime@ Range[10^3], PrimeQ[# + 12] &] (* Michael De Vlieger, Feb 25 2017 *)

A101232 First occurrence of prime gap 10^n.

Original entry on oeis.org

2, 139, 396733, 22439962446379651

Offset: 0

Lekraj Beedassy, Dec 16 2004

Schiffman, Prime Curios! 22439962446379651

Cf. A000230, A062529.

A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

Original entry on oeis.org

2, 4, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, 6469693230, 200560490130, 7420738134810, 304250263527210, 13082761331670030, 614889782588491410, 32589158477190044730, 1922760350154212639070

Offset: 1

Lekraj Beedassy, Aug 10 2006

If n > 2, then a(n) = product of n-1 consecutive distinct prime divisors. E.g. a(5)=210, the product of 4 consecutive and distinct prime divisors, 2,3,5,7. - Enoch Haga, Dec 08 2007
From Bill McEachen, Jul 10 2022: (Start)
Rather than have code merely generating the conjectured values, one can compare values of sequence terms at the same position n. Specifically, locate new maximums where (p,p+even) are both prime, where even=2,4,6,8,... and the datum set is taken with even=4. A new maximum implies a new jumping champion.
Doing this produces the terms 2,4,6,30,210,2310,30030,.... Looking at the plot of a(n) ratio for gap=2/gap=6, the value changes VERY slowly, and is 2.14 after 50 million terms (one can see the trend via Plot 2 of A001359 vs A023201 (3rd option seqA/seqB vs n). The ratio for gap=4/gap=2 ~ 1, implying they are equally frequent. (End)

R. P. Brent, The First Occurrence of Large Gaps Between Successive Primes
R. P. Brent, The distribution of small gaps between successive primes
R. P. Brent, The first occurrence of certain large prime gaps
C. K. Caldwell, The Prime Glossary, gaps between primes
C. K. Caldwell, The Prime Glossary, Jumping champion
S. Funkhouser, D. A. Goldston, D. Sengupta, and J. Sengupta, Prime Difference Champions, arXiv:1612.02938 [math.NT], 2016.
D. A. Goldston and A. H. Ledoan, Jumping champions and gaps between consecutive primes, Oct 15, 2009. [From _Jonathan Vos Post_, Oct 17 2009]
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions
A. M. Odlyzko, M. Rubinstein, and M. Wolf, CHANCE News 10.02, 10. Jumping champions in the world of primes
A. M. Odlyzko, M. Rubinstein, and M. Wolf, Jumping Champions, Experiment. Math. 8(2): 107-118 (1999).
Tomás Oliveira e Silva, Gaps between consecutive primes
Ian Stewart, Jumping Champions, Scientific American, Vol. 283, No. 6 (2000), pp. 106-107; Wayback Machine link.
Eric Weisstein's World of Mathematics, Jumping Champion

Cf. A087103, A087104, A001223, A000230, A001632, A038664, A086977-A086980, A085237, A005250, A053686, A054587, A093737-A093753, A093972-A093984.

2,4,Table[Product[Prime[k],{k,1,n-1}],{n,3,30}]

print1("2, 4");t=2;forprime(p=3,97,print1(", ",t*=p)) \\ Charles R Greathouse IV, Jun 11 2011

Consists of 4 and the primorials (A002110).

a(1) = 2, a(2) = 4, a(3) = 6, a(n+1)/a(n) = Prime[n] for n>2.

Corrected and extended by Alexander Adamchuk, Aug 11 2006

Definition corrected and clarified by Jonathan Sondow, Aug 16 2011

A144103 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,2) - p = 2*n, or -1 if no such prime exists.

Original entry on oeis.org

-1, 3, 5, 23, 19, 47, 83, 211, 109, 317, 619, 199, 1373, 1123, 1627, 4751, 2557, 3413, 4289, 1321, 2161, 2477, 7963, 5591, 9551, 17239, 15823, 14087, 19603, 34963, 36389, 33223, 24251, 35603, 43321, 19609, 134507, 31393, 136999, 31397, 38461, 107357

Offset: 1

T. D. Noe, Sep 11 2008

sign

p and p+2n are primes and there is one prime in the range p+1 to p+2n-1.

a(n) is the prime for which 2n+2 first occurs in A031131.

Martin Raab, Table of n, a(n) for n = 1..469 (terms 1..342 from Robert G. Wilson v)
Mersenne Forum, Gaps between non-consecutive primes

Cf. A031131.

A000230 is an analogous sequence based on N(p,1). - N. J. A. Sloane, Nov 07 2020

nn=51; t=Table[0,{nn}]; t[[1]]=-1; cnt=1; n=1; While[cntHarvey P. Dale, Jun 26 2017 *)

Definition edited by N. J. A. Sloane, Nov 07 2020

A298366 Even numbers n such that n-5 and n-3 are both composite.

Original entry on oeis.org

30, 38, 54, 60, 68, 80, 90, 96, 98, 120, 122, 124, 126, 128, 138, 146, 148, 150, 158, 164, 174, 180, 188, 190, 192, 206, 208, 210, 212, 218, 220, 222, 224, 240, 248, 250, 252, 258, 264, 270, 278, 290, 292, 294, 300, 302, 304, 306, 308, 324, 326, 328, 330, 332, 338, 344, 346, 348, 360, 366, 368, 374, 380

Offset: 1

David James Sycamore, Jan 17 2018

nonn

The sequence displays runs of consecutive even integers, whose frequency and length are related to gaps between successive primes local to these numbers. Where primes are rare (large gaps), the runs of consecutive even integers are longer (run length proportional to gap size). Let p < q be consecutive primes such that g = q-p >= 6. A string of r consecutive terms differing by 2 will start at p+7, and continue to q+1, where r = (g-4)/2. Thus at prime gap 8 a string of 2 consecutive terms differing by 2 will occur, at gap 10 there will be 3, and at gap 30 there will be 13; and so on. As the gap size increases by 2 so the run length of consecutive even terms increases by 1. The first occurrence of run length m occurs at the term corresponding to 7 + A000230(m/2).
The terms in this sequence, combined with those in A297925 and A298252 form a partition of A005843(n); n >= 3; (nonnegative even numbers >= 6). This is because any even integer n >= 6 satisfies either: (i). n-3 is prime, (ii). n-5 is prime and n-3 is composite, or (iii). both n-5 and n-3 are composite.
For any n >= 1, A056240(a(n)) = A298615(n).

			30 is included because 30-5 = 25, and 30-3 = 27; both composite, and 30 is the smallest even number with this property, hence a(1)=30. Also, A056240(a(1)) = A056240(30) = 161 = A298615(1). 24 is not included because although 24 - 3 = 21, composite; 24 - 5 = 19, prime. 210 is in this sequence, since 205 and 207 are both composite. 113 is the first prime to have a gap 14 ahead of it. Therefore we would expect a run of (14 - 4)/2 = 5 consecutive terms to start at 7 + A000230(7) = 113 + 7 = 120; thus: 120,122,124,126,128. Likewise the first occurrence of run length 7 occurs at gap m = 2*7 + 4 = 18, namely the term corresponding to 7 + A000230(9) = 523 + 7 = 530; thus: 530,532,534,536,538,540,542.

Vincenzo Librandi, Table of n, a(n) for n = 1..7000

Cf. A297925, A298252, A005843, A000230.

[2*n: n in [8..200] | not IsPrime(2*n-5) and not IsPrime(2*n-3)]; // Vincenzo Librandi, Nov 16 2018

N:=300:
for n from 8 to N by 2 do
if not isprime(n-5) and not isprime(n-3) then print(n);
end if
end do

Rest[2 Select[Range[250], !PrimeQ[2 # - 5] && !PrimeQ[2 # - 3] &]] (* Vincenzo Librandi, Nov 16 2018 *)
Select[Range[2,400,2],AllTrue[#-{3,5},CompositeQ]&] (* Harvey P. Dale, Jul 01 2025 *)

select( is_A298366(n)=!(isprime(n-5)||isprime(n-3)||bitand(n,1)||n<9), [5..200]*2) \\ Last 2 conditions aren't needed if n > 4 and even. - M. F. Hasler, Nov 19 2018 and Apr 07 2020 after edit by Michel Marcus, Apr 04 2020

a(n) = A061673(n) + 4 = A269345(n) + 5. - M. F. Hasler, Nov 19 2018

A046789 Lower prime of the second gap of 2n between primes.

Original entry on oeis.org

5, 13, 31, 359, 181, 211, 293, 1933, 1069, 1637, 1951, 2179, 3137, 3271, 4831, 10799, 8467, 12853, 38393, 20809, 34981, 36389, 91033, 73189, 45893, 25471, 40289, 153191, 58831, 102701, 190409, 107377, 221327, 290249, 175141, 360091

Offset: 1

Jud McCranie

nonn

			The second prime gap of 4 is at 13 to 17, so a(2)=13.

Enoch Haga, Exploring Prime Numbers on Your PC, 2nd edition, 1998, ISBN 1-885794-16-9, Table 4, pages 38-39.

Robert G. Wilson v, Table of n, a(n) for n = 1..257

Cf. A000230, A046728.

Flatten[Table[First /@ Take[Select[Partition[Prime[Range[32000]], 2, 1], Differences[#] == {2 n} &], {2}], {n, 36}]] (* Jayanta Basu, Jun 28 2013 *)
tt[] := -2; p0 = NextPrime@2; p1 = NextPrime@p0; While[p0 < 25000000, diff = (p1 - p0)/2; If[tt[diff] == -1, tt[diff] = p0]; If[tt[diff] == -2, tt[diff] = -1]; {p0, p1} = {p1, NextPrime@p1}]; tt@# & /@ Range@77 (* _Robert G. Wilson v, Nov 26 2020 *)

a(n) = A000230(n) + A046728(n). - Robert G. Wilson v, Nov 26 2020

A086979 Increasing peaks in the prime gap sequence A038664.

Original entry on oeis.org

46, 282, 738, 3302, 7970, 8028, 14862, 15783, 34202, 44773, 44903, 85787, 110224, 165326, 402884, 460883, 474029, 786922, 887313, 2959782, 4875380, 8321465, 9330121, 20226285, 45808557, 92276646, 114867712, 201745031, 265878477

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is Pi(p_k), the number of primes up to and including p_k, where p_k is the initial prime of a prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime and the next even gap g+2 also occurs earlier.

			282 is in this list because the 282nd prime is 1831, the next prime is 1847, giving a prime gap of 16. All even gaps less than 16 occur before this (for smaller primes) and the next even gap, 18, also occurs earlier.

P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 144.

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

Cf. A000230, A001223, A001632, A002386, A038664, A086977, A086978, A086980.

cp's OEIS Frontend

A140791 First occurrence of prime gap 10*n.

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A058193 Smallest prime p such that there is a gap of 6n between p and the next prime.

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A062529 Smallest prime p such that there is a gap of 2^n between p and the next prime.

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A101232 First occurrence of prime gap 10^n.

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A121069 Conjectured sequence for jumping champions greater than 1 (most common prime gaps up to x, for some x).

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A144103 Let N(p,i) denote the result of applying "nextprime" i times to p; a(n) = smallest prime p such that N(p,2) - p = 2*n, or -1 if no such prime exists.

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A298366 Even numbers n such that n-5 and n-3 are both composite.

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Magma