A002386 -id:A002386 - cp's OEIS frontend

A182876 Ramanujan primes that begin a record gap to the next Ramanujan prime.

2, 17, 71, 181, 503, 2531, 2909, 3067, 5273, 7001, 7741, 9887, 11587, 33911, 55339, 174917, 225961, 345041, 360289, 534883, 2492311, 5409337, 20099089, 31531063, 39706123, 49444063, 80925371, 141833603, 180881749, 197920843, 212583797, 658046911, 777899149, 1183597123, 1595746589, 2897211971

Offset: 1

T. D. Noe, Dec 09 2010

nonn

See A182877 for the length of the gap.

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

Cf. A002386 (primes beginning a record gap)

A214924 Number of primes <= A214756(n).

Original entry on oeis.org

1, 1, 1, 7, 20, 28, 96, 152, 185, 212, 1179, 1829, 2217, 3382, 14350, 30780, 31528, 40929, 103498, 104047, 149674, 325845, 1094396, 1319933, 2850163, 6957867, 10539421, 10655453

Offset: 1

John W. Nicholson, Jul 29 2012

nonn

a(n) = pi(A214756(n)).

			A214756(5) = 71, so a(5) = primepi(A214756(5)) = primepi(71) = 20.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214925, A214926, A214757.

a(n) = A000217(A214756(n))

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A214925 Number of primes <= A214757(n).

Original entry on oeis.org

5, 5, 5, 10, 25, 31, 104, 159, 190, 219, 1186, 1832, 2227, 3388, 14358, 30804, 31547, 40935, 103522, 104072, 149690, 325853, 1094426, 1319950, 2850175, 6957880, 10539433, 10655464

Offset: 1

John W. Nicholson, Aug 06 2012

			A214757(4) = 29, so a(4) = primepi(A214757(4)) = primepi(29) = 10.

Cf. A000101, A104272, A214756, A001223.

Cf. A002386, A214924, A214926, A214757.

a(n) = pi(A214757(n)) = A000217(A214757(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A214926 Difference A214925(n) - A214924(n), prime count between Ramanujan primes bounding maximal gap primes.

Original entry on oeis.org

4, 4, 4, 3, 5, 3, 8, 7, 5, 7, 7, 3, 10, 6, 8, 24, 19, 6, 24, 25, 16, 8, 30, 17, 12, 13, 12, 11

Offset: 1

John W. Nicholson, Aug 06 2012

nonn

Conjecture: For every n > 0, a(n) > 1.

Let rho(m) = A179196(m), for any n, let m be an integer such that p_(rho(m)) <= p_n and p_(n+1) <= p_(rho(m+1)), then rho(m) <= n < n + 1 <= rho(m + 1), therefore A001223(n) = p_(n+1) - p_n <= p_rho(m+1) - p_rho(m) = A182873(m). For all rho(m) = A179196(m), A001223(rho(m)) < A165959(m). (Comment copied from A001223). John W. Nicholson, Nov 17 2013

			a(4) = pi(A214757(4)) - pi(A214756(4)) = 10 - 7 = 3

Cf. A214924, A214925, A214756, A214757.

Cf. A000101, A104272, A001223, A002386.

a(n) = pi(A214757(n)) - pi(A214756(n)).

a(n) = rho(A214757(n)) - rho(A214756(n)).

Extension to a(28) added by John W. Nicholson, Nov 11 2013

A249566 Numbers n such that A182134(n) = 4, i.e., there exist exactly four primes p with prime(n) < p < prime(n)^(1+1/n).

Original entry on oeis.org

17, 19, 24, 26, 32, 33, 35, 36, 37, 38, 40, 42, 43, 47, 50, 51, 52, 58, 62, 63, 64, 76, 77, 78, 79, 90, 91, 93, 95, 121, 123, 124, 125, 126, 134, 135, 137, 150, 153, 185, 186, 187, 188, 189, 201, 203, 213, 218, 219, 238, 239, 259, 263, 278, 279, 289, 293

Offset: 1

Robert Price, Nov 01 2014

nonn

See A246782 for a more complete description of this sequence.
a(1136) > 10^12.
It is interesting that three consecutive integers n = 20004097201301075, n + 1 and n + 2 are in the sequence. Conjecture: The sequence is infinite. - Farideh Firoozbakht, Nov 01 2014

Robert Price, Table of n, a(n) for n = 1..1135
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015
Wikipedia, Firoozbakht's conjecture

Cf. A000040, A002386, A005669, A182134, A246781, A246782.

a249566 n = a249566_list !! (n-1)
a249566_list = filter ((== 4) . a182134) [1..]
-- Reinhard Zumkeller, Nov 17 2014

np[n_]:=(a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a+1,b], PrimeQ]]); Do[If[np[n] == 4,Print[n]], {n, 293}]
np[n_]:=(a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a+1,b], PrimeQ]]); Select[Range[293], np[#]==4&] (* Farideh Firoozbakht, Nov 01 2014 *)

for(n=1,9e9,primepi(prime(n)^(1+1/n))-n==4&&print1(n",")) \\ M. F. Hasler, Nov 03 2014

A077019 a(n) is the smallest number for which the prime distance A051699 is equal to n.

Original entry on oeis.org

2, 1, 0, 26, 93, 118, 119, 120, 531, 532, 897, 1140, 1339, 1340, 1341, 1342, 1343, 1344, 9569, 15702, 15703, 15704, 15705, 19632, 19633, 19634, 19635, 31424, 31425, 31426, 31427, 31428, 31429, 31430, 31431, 31432, 31433, 155958, 155959, 155960, 155961, 155962, 155963, 155964

Offset: 0

Eric W. Weisstein, Oct 17 2002

nonn

David A. Corneth, Table of n, a(n) for n = 0..775 (using b-file from A002386)
Eric Weisstein's World of Mathematics, Prime Distance

Cf. A002386, A051699.

d(n) = if(n<1, 2*(n==0), min(nextprime(n)-n, n-precprime(n))); \\ A051699
a(n) = my(k=0); while (d(k) != n, k++); k; \\ Michel Marcus, Aug 21 2019

More terms from Michel Marcus, Aug 21 2019

A086978 Increasing peaks in the prime gap sequence A001632.

Original entry on oeis.org

211, 1847, 5623, 30631, 81509, 82129, 162209, 173429, 404671, 542683, 544367, 1101071, 1444411, 2238931, 5845309, 6752747, 6958801, 11981587, 13626407, 49269739, 83751287, 147684323, 166726561, 378044179, 895858267, 1872852203

Offset: 1

Harry J. Smith, Jul 26 2003

nonn

a(n) is the larger 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.

			1847 is in this list because the previous prime is 1831, 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, A086977, A086979, A086980, A002386.

A087770 "Lonely primes": those primes that are locally maximally isolated from the nearest other primes. The differences between each lonely prime and the immediately preceding prime and following primes are both greater than the corresponding differences for all lonely primes earlier in the sequence.

Original entry on oeis.org

2, 3, 7, 23, 89, 211, 1847, 2179, 14107, 33247, 38501, 58831, 268343, 1272749, 2198981, 10938023, 72546283, 162821917, 325737821, 2888688863, 6613941601, 11179888193, 24016237123, 96155166493, 179474021633, 215686840471, 633880576177, 1480975873513, 9156364643509

Offset: 1

Walter Carlini, Oct 03 2003

nonn

The concept of "lonely prime" is similar to that of maximal prime gaps since lonely primes are increasingly distant from each other.

See A023186 for another version of this sequence, which only requires increasing the minimum of the two gaps to the neighbors. The definition from A023186 seems to be the more common variant. - Hugo Pfoertner, Dec 17 2019

			a(0) = 2.
a(1) = 3 because 3 - 2 = 1 and 5 - 3 = 2.
a(2) = 7 because 7 - 5 = 2 (and 2 > 3 - 2) and 11 - 7 = 4 (and 4 > 5 - 3).
a(3) = 23 because 23 - 19 = 4 ( 23 - 19 > 7 - 5) and 29 - 23 = 6 (29 - 23 > 11 - 7).
a(4) = 89 because 89 - 83 = 6 > 23 - 19 and 97 - 89 = 8 > 29 - 23.
Note, for example, that 53 is not a lonely prime because 53 - 47 = 6, which is > 23 - 19 however 59 - 53 = 6, which is not > 29 - 23.

Cf. A000230, A002386, A023186.

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; p = 2; q = 2; r = 3; d = e = 0; Do[ While[ q - p <= d || r - q <= e, p = q; q = r; r = NextPrim[r]]; Print[q]; d = Max[q - p, d]; e = Max[r - q, e]; p = q; q = r; r = NextPrim[r], {n, 1, 40}] (* Robert G. Wilson v *)

Corrected and extended by Ray Chandler, Oct 06 2003
Offset changed and a(21)-a(27) from Hugo Pfoertner, Dec 17 2019
a(28)-a(29) from Giovanni Resta, Dec 17 2019

A113403 Primes p in prime quadruplets (p,p+2,p+6,p+8) at the end of maximal gaps in A113404.

Original entry on oeis.org

11, 101, 821, 1481, 3251, 5651, 9431, 31721, 43781, 97841, 135461, 187631, 326141, 768191, 1440581, 1508621, 3047411, 3798071, 5146481, 5610461, 9020981, 17301041, 22030271, 47774891, 66885851, 76562021, 87797861, 122231111, 132842111, 204651611, 628641701, 1749878981

Offset: 1

Bernardo Boncompagni, Oct 28 2005

nonn

Prime quadruplets (p, p+2, p+6, p+8) are densest permissible constellations of four primes. Record (maximal) gaps between prime quadruplets are listed in A113404; see further comments there.

			Record gaps between prime quadruplets are as follows (arXiv:1309.4053, Table 4):
Initial primes: Max gap
.....5......11........6
....11.....101.......90
...191.....821......630
...821....1481......660
..2081....3251.....1170
..3461....5651.....2190
..5651....9431.....3780
...
The left column is A229907. The middle column is A113403 (this sequence); the right column is A113404.

Jud McCranie and Alexei Kourbatov, Table of n, a(n) for n = 1..71
Tony Forbes and Norman Luhn, Prime k-tuplets.
Alexei Kourbatov, Maximal gaps between prime k-tuples.
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053, 2013.
Eric Weisstein's World of Mathematics, k-Tuple Conjecture.

Record gaps are given in A113404. Cf. A007530, A002386.

A132470 Smallest number at distance exactly 3n from nearest prime.

Original entry on oeis.org

2, 26, 119, 532, 1339, 1342, 9569, 15704, 19633, 31424, 31427, 31430, 31433, 155960, 155963, 360698, 360701, 370312, 370315, 492170, 1357261, 1357264, 1357267, 2010802, 2010805, 4652428, 17051785, 17051788, 17051791, 17051794, 17051797, 20831416, 20831419, 20831422

Offset: 0

Jonathan Vos Post, Sep 03 2007

nonn

Let f(m)= A051699(m) = exact distance from m to its closest prime (including m itself). Then a(n) = min { m : f(m) = 3n}. - R. J. Mathar, Nov 18 2007

This sequence can be derived from the record prime gap sequences A002386 and A005250. In particular, for n > 0, a(n) = A002386(k) + 3*n where k is the least index such that A005250(k) >= 3*n. - Andrew Howroyd, Jan 04 2020

			a(3)=532 where 532+3*3 is prime and all numbers below 532 have a distance smaller or larger than 3n=9 to their nearest primes and there is no prime within a distance of 8 to 532.

Andrew Howroyd, Table of n, a(n) for n = 0..258

Cf. A002386, A005250, A051699, A051728.

A051699 := proc(m) if isprime(m) then 0 ; elif m <= 2 then op(m+1,[2,1]) ; else min(nextprime(m)-m,m-prevprime(m)) ; fi ; end: A132470 := proc(n) local m ; if n = 0 then RETURN(2); else for m from 0 do if A051699(m) = 3 * n then RETURN(m) ; fi ; od: fi ; end: seq(A132470(n),n=0..18) ; # R. J. Mathar, Nov 18 2007

terms = 34;
gaps = Cases[Import["https://oeis.org/A002386/b002386.txt", "Table"], {, }][[;; terms, 2]];
w[n_] := (NextPrime[gaps[[n]] + 1] - gaps[[n]])/6 // Floor;
k = 1; a[0] = 2;
For[n = 1, n <= terms, n++, While[w[k] < n, k++]; a[n] = gaps[[k]] + 3n];
a /@ Range[0, terms-1] (* Jean-François Alcover, Apr 09 2020, after Andrew Howroyd *)

\\ here R(gaps) wants prefix of A002386 as vector.
aA002386(lim)={my(L=List(),q=2,g=0); forprime(p=3, lim, if(p-q>g, listput(L,q); g=p-q); q=p); Vec(L)}
R(gaps)={my(w=vector(#gaps, n, nextprime(gaps[n]+1) - gaps[n])\6, r=vector(w[#w]+1), k=1); r[1]=2; for(n=1, w[#w], while(w[k]A002386(10^7))} \\ Andrew Howroyd, Jan 04 2020

a(n) = min {m : A051699(m) = 3n}. - R. J. Mathar, Nov 18 2007

Corrected by Dean Hickerson, Sep 05 2007
Both this sequence and A051728 should be checked. There are two possibilities for confusion in each case. In defining f(m), does one allow or exclude m itself, in case m is a prime? In defining a(n), does one require (here) that f(m) = 3n or only that >= 3n, or (in A051728) that f(m) = 2n or only >= 2n? Probably there should be several sequences, to include all the possibilities in each case. - N. J. A. Sloane, Nov 18 2007. Added Nov 20 2007: R. J. Mathar has now clarified the definition of the present sequence.
Corrected and extended by R. J. Mathar, Nov 18 2007
Terms a(19) and beyond from Andrew Howroyd, Jan 04 2020

cp's OEIS Frontend

A182876 Ramanujan primes that begin a record gap to the next Ramanujan prime.

Views

Author

Keywords

Comments

Links

Crossrefs

A214924 Number of primes <= A214756(n).

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A214925 Number of primes <= A214757(n).

Views

Author

Keywords

Examples

Crossrefs

Formula

Extensions

A214926 Difference A214925(n) - A214924(n), prime count between Ramanujan primes bounding maximal gap primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A249566 Numbers n such that A182134(n) = 4, i.e., there exist exactly four primes p with prime(n) < p < prime(n)^(1+1/n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

A077019 a(n) is the smallest number for which the prime distance A051699 is equal to n.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Extensions

A086978 Increasing peaks in the prime gap sequence A001632.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A113403 Primes p in prime quadruplets (p,p+2,p+6,p+8) at the end of maximal gaps in A113404.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs