A002386 -id:A002386 - cp's OEIS frontend

A086977 Increasing peaks in the prime gap sequence A000230.

199, 1831, 5591, 30593, 81463, 82073, 162143, 173359, 404597, 542603, 544279, 1100977, 1444309, 2238823, 5845193, 6752623, 6958667, 11981443, 13626257, 49269581, 83751121, 147684137, 166726367, 378043979, 895858039, 1872851947

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the smaller of the two consecutive primes having a late occurring prime gap g = p_k+1 - p_k. All even gaps smaller than g occur at a smaller prime. Also, the next even gap g+2 also occurs earlier.

			1831 is in this list because 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, A038664, A086978, A086979, A086980, A002386.

lst={};b=max=2;Do[a=2;While[NextPrime@a-a!=2n,a=NextPrime@a];If[a=max,AppendTo[lst,b]];b=a;If[b>max,max=b],{n,40}];lst (* Giorgos Kalogeropoulos, Aug 18 2021 *)

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.

A086980 Late occurring prime gaps in the prime gap sequence A001223.

Original entry on oeis.org

12, 16, 32, 38, 46, 56, 66, 70, 74, 80, 88, 94, 102, 108, 116, 124, 134, 144, 150, 158, 166, 186, 194, 200, 228, 256, 264, 278, 294, 298, 316, 328, 334, 362, 370, 388, 422, 436, 442, 452, 466, 472, 482, 488, 510, 520, 536, 568, 576, 580, 590, 608, 628, 632

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the gap g = p_k+1 - p_k between consecutive primes with all even gaps smaller than g occurring at a smaller prime and the next even gap g+2 also occurring earlier.

			16 is in this list because the first time a prime gap of 16 occurs is between consecutive primes 1831 and 1847. All even prime gaps less than 16 occur for a smaller prime. The next even prime gap of 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, A038664, A086977, A086978, A086979, A002386.

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

A124129 Primes p for which there are no primes between p and p+sqrt(p).

Original entry on oeis.org

3, 7, 13, 23, 31, 113

Offset: 1

Rémi Eismann, Dec 10 2006

Conjecture: there are no other terms.
The finiteness of this sequence would follow from Cramer's conjecture that lim sup (p(n+1)-p(n))/log(p(n))^2 = 1. - Dean Hickerson, Dec 12 2006
The finiteness of this sequence would imply that, for every sufficiently large positive integer n, there is a prime between n^2 and (n+1)^2. Except for the "sufficiently large", that's Legendre's conjecture, which is still unproved. - Dean Hickerson, Dec 12 2006
There are no other terms less than 218034721194214273 (assuming that the extended table of terms in A002386 is correct). - Dean Hickerson, Dec 12 2006
The evidence suggests that for any k, the number of primes with p < gap(p)^k is finite (this sequence being the special case k = 2), where gap(p) is the difference between p and the next prime. - David W. Wilson, Dec 13 2006
Primes for which sqrt(A000040(n)) < A001223(n).
Also primes p(n) for which the remainder of the division of p(n)^2 by p(n+1) is different from the remainder of the division of p(n+1)^2 by p(n).

			a(1) = 3 because sqrt(3) < 2. a(6) = 113 because sqrt(113) < 14.

A. Granville, Harald Cramér and the distribution of prime numbers, Scandinavian Actuarial Journal, Volume 1995, 1995 - Issue 1.
Matt Visser, Strong version of Andrica's conjecture, arXiv:1812.02762 [math.NT], 2018.

Cf. A000040, A001223, A002386.

Select[ Prime@ Range@100, PrimePi[ # + Sqrt@# ] - PrimePi@# == 0 &] (* Robert G. Wilson v, Dec 18 2006 *)

A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

2, 6, 14, 190, 194, 200, 306, 462, 468, 474, 478, 490, 560, 1208, 1890, 1938, 23716, 23850, 25226, 25834, 25968, 26642, 26650, 26998, 48316, 311888, 311922, 313946, 331540, 331762, 331782, 377078, 377518, 377666, 377674, 377748, 378422, 378428

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 170 terms < 10^8.

Being prime(n) = 1 + Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 odd and, prime(n) = (1 + Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 even.

			1 + ( -A001223(1)) = 1+(-1) = 0, hence 2 is a term.
1 + ( -A001223(1) + A001223(3) - A001223(5)) = 1+(-1+2-2) = 0, hence 6 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1+S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A130643 Numbers n such that 1 - Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

Original entry on oeis.org

4, 8, 12, 22, 38, 302, 308, 464, 472, 476, 1186, 1884, 2006, 2026, 2106, 23636, 23656, 23698, 25984, 25990, 26706, 26924, 27000, 311914, 311938, 313866, 313880, 331676, 332002, 332676, 377102, 377634, 377670, 379026, 379090, 379108, 387618, 389076

Offset: 1

Manuel Valdivia, Jun 20 2007

Sequence has 177 terms < 10^8.

Being prime(n) = 1 - Sum{k=1..n-1}A000040(k)*(-1)^Floor(k/2), for n/2 even and, prime(n) = (1 - Sum{k=1..n- 1}A000040(k)*(-1)^Floor(k/2))*(-1), for n/2 odd.

			1 - ( -A001223(1) + A001223(3)) = 1-(-1+2) = 0, hence 4 is a term.
1 - ( -A001223(1) + A001223(3) - A001223(5) + A001223(7)) = 1-(-1+2-2+2) = 0, hence 8 is a term.

Cf. A127596, A128039, A001223, A000101, A002386.

S=0; a=0; Do[S=S+(Prime[2*k]-Prime[2*k-1])*(-1)^k; If[1-S==0, a++; Print[a, " ", 2*k]], {k, 1, 10^8, 1}]

A182315 Primes prime(n) such that prime(n+1) - prime(n) > log(n)^2.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 31, 113, 1327, 31397, 370261, 492113, 2010733, 20831323, 25056082087, 42652618343, 2614941710599, 19581334192423

Offset: 1

Thomas Ordowski, Apr 24 2012

Using terms of A002386, a(19) is probably 218209405436543. - T. D. Noe, Apr 24 2012

Subsequence of A211073.

Cf. A002386, A134266.

t = {}; Do[If[Prime[n + 1] - Prime[n] > Log[n]^2, AppendTo[t, Prime[n]]], {n, 10000}]; t (* T. D. Noe, Apr 24 2012 *)

n=0;G=1;p=2;forprime(q=3,1e8,n++;if(q-p>=G&&q-p>log(n)^2, G=ceil(log(n)^2);print1(p", "));p=q) \\ Charles R Greathouse IV, Apr 24 2012

a(13)-a(16) from Charles R Greathouse IV, Apr 24 2012
a(17) from Charles R Greathouse IV, Apr 26 2012
a(18) from Charles R Greathouse IV, May 06 2012

cp's OEIS Frontend

A086977 Increasing peaks in the prime gap sequence A000230.

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A086979 Increasing peaks in the prime gap sequence A038664.

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A086980 Late occurring prime gaps in the prime gap sequence A001223.

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

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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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A124129 Primes p for which there are no primes between p and p+sqrt(p).

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A130642 Numbers n such that 1 + Sum{k=1..n/2} A001223(2k-1)*(-1)^k = 0.

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