A134266 -id:A134266 - cp's OEIS frontend

A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

2, 3, 7, 23, 89, 113, 523, 887, 1129, 1327, 9551, 15683, 19609, 31397, 155921, 360653, 370261, 492113, 1349533, 1357201, 2010733, 4652353, 17051707, 20831323, 47326693, 122164747, 189695659, 191912783, 387096133, 436273009, 1294268491

Offset: 1

N. J. A. Sloane

See the links by Jens Kruse Andersen et al. for very large gaps.

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 133.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 3, Sect 6.1, Table 1.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 14.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Brian Kehrig, Table of n, a(n) for n = 1..83 (first 75 terms from M. F. Hasler and N. J. A. Sloane, terms n = 76..77 added by Charles R Greathouse IV)
R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
R. K. Guy, Letter to N. J. A. Sloane, 1987
Lutz Kämmerer, A fast probabilistic component-by-component construction of exactly integrating rank-1 lattices and applications, arXiv:2012.14263 [math.NA], 2020.
Jens Kruse Andersen and Norman Luhn, Record Prime Gaps
Alexei Kourbatov, Tables of record gaps between prime constellations, arXiv preprint arXiv:1309.4053 [math.NT], 2013.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015; and J. Int. Seq. 18 (2015) #15.11.2.
Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Thomas R. Nicely, New maximal prime gaps and first occurrences, Math. Comput. 68,227 (1999) 1311-1315.
Tomás Oliveira e Silva, Gaps between consecutive primes
D. Shanks, On maximal gaps between successive primes, Math. Comp., 18 (1964), 646-651.
Matt Visser, Verifying the Firoozbakht, Nicholson, and Farhadian conjectures up to the 81st maximal prime gap, arXiv:1904.00499 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Prime Gaps
Wikipedia, Prime gap
Robert G. Wilson v, Notes (no date)
J. Young and A. Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.
Index entries for primes, gaps between

Cf. A000040, A001223, A000101 (upper ends), A005250 (record gaps), A000230, A111870, A111943.
Cf. A005669, A134266, A070866.
See also A205827(n) = A000040(A214935(n)), A182514(n) = A000040(A241540(n)).

s = {2}; gm = 1; Do[p = Prime[n]; g = Prime[n + 1] - p; If[g > gm, Print[p]; AppendTo[s, p]; gm = g], {n, 2, 1000000}]; s   (* Jean-François Alcover, Mar 31 2011 *)
Module[{nn=10^7,pr,df},pr=Prime[Range[nn]];df=Differences[pr];DeleteDuplicates[ Thread[ {Most[ pr],df}],GreaterEqual[#1[[2]],#2[[2]]]&]][[All,1]] (* The program generates the first 26 terms of the sequence. *) (* Harvey P. Dale, Sep 24 2022 *)

a(n)=local(p,g);if(n<2,2*(n>0),p=a(n-1);g=nextprime(p+1)-p;while(p=nextprime(p+1),if(nextprime(p+1)-p>g,break));p) /* Michael Somos, Feb 07 2004 */

p=q=2;g=0;until( g<(q=nextprime(1+p=q))-p && print1(q-g=q-p,","),) \\ M. F. Hasler, Dec 13 2007

a(n) = A000101(n) - A005250(n) = A008950(n-1) - 1. - M. F. Hasler, Dec 13 2007
A000720(a(n)) = A005669(n).
a(n) = A000040(A005669(n)). - M. F. Hasler, Apr 26 2014

Definition clarified by Harvey P. Dale, Sep 24 2022

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

A348178 The list of all prime numbers is split into sublists with the 1st sublist L_1 = {2} and n-th sublist L_n = {p_1, p_2, ..., p_m}. a(n) is the largest m such that the maximum prime gap in L_n is < p_1 - prevprime(p_1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 4, 1, 2, 3, 2, 1, 6, 32, 4, 33, 55, 35, 28, 842, 124, 349, 131, 168, 394, 585, 575, 10972, 14683, 1762, 743, 9388, 62587, 551, 14434, 31184, 176163, 407736, 249427, 111406, 225524, 1530229, 4107702, 3581556, 116030, 10028870, 2065372

Offset: 1

Ya-Ping Lu, Oct 05 2021

nonn

The last prime in the n-th sublist is A134266(n). The gap between the n-th and (n+1)-th sublists is A085237(n).

Cf. A085237, A134266, A348168.

from sympy import nextprime
L = [2]
for n in range(1, 50):
    print(len(L), end = ', ')
    p0 = L[-1]; p1 = nextprime(p0); g0 = p1 - p0; M = [p1]; p = nextprime(p1)
    while p - p1 < g0: M.append(p); p1 = p; p = nextprime(p)
    L = M

A085500 Indices of primes where nondecreasing gaps occur.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 11, 15, 16, 18, 21, 23, 24, 30, 62, 66, 99, 154, 189, 217, 1059, 1183, 1532, 1663, 1831, 2225, 2810, 3385, 14357, 29040, 30802, 31545, 40933, 103520, 104071, 118505, 149689, 325852, 733588, 983015, 1094421, 1319945, 2850174, 6957876, 10539432, 10655462

Offset: 1

Farideh Firoozbakht, Aug 15 2003

nonn

A005669 is a subsequence of this sequence.

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A8, pp. 31-39.

Jeff Young and Aaron Potler, First occurrence prime gaps, Math. Comp., 52 (1989), 221-224.

Cf. A085237, A005669, A005250, A134266.

See also A214935(n) = A000720(A205827(n)), A241540(n) = A000720(A182514(n)).

f[n_] := Prime[n+1]-Prime[n]; v1={}; v2={}; Do[If[f[n]>=If[n==1, 1, Last[v2]], v=n; v1=Append[v1, n]; v2=Append[v2, f[v]]; Print[v1]], {n, 105000000}]

a(n) = A000720(A134266(n)). - M. F. Hasler, Apr 26 2014

a(45)-a(47) from Amiram Eldar, Sep 05 2024

A361823 a(1) = 3; thereafter, a(n+1) is the smallest prime p such that p - prevprime(p) >= a(n) - prevprime(a(n)).

Original entry on oeis.org

3, 5, 7, 11, 17, 23, 29, 37, 53, 59, 67, 79, 89, 97, 127, 307, 331, 541, 907, 1151, 1361, 8501, 9587, 12889, 14143, 15727, 19661, 25523, 31469, 156007, 338119, 360749, 370373, 492227, 1349651, 1357333, 1562051, 2010881, 4652507, 11114087, 15204131, 17051887

Offset: 1

Ya-Ping Lu, Mar 25 2023

nonn

a(n) is the leading prime in the (n+1)-th prime sublist defined in A348178.

Cf. A001223, A070866, A134266, A348178.

a361823(upto) = {my(pp=2, gap=1); forprime (p=3, upto, my(g=p-pp);if(g>=gap, print1(p,", "); gap=g); pp=p)};
a361823(20000000) \\ Hugo Pfoertner, Apr 03 2023

from sympy import nextprime; q = 2; g = 0
while q < 20000000:
    p = nextprime(q); d = p - q
    if d >= g: print(p, end = ', '); g = d
    q = p

a(n) = nextprime(A134266(n)). - Michel Marcus, Mar 30 2023

cp's OEIS Frontend

A002386 Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.

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

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A348178 The list of all prime numbers is split into sublists with the 1st sublist L_1 = {2} and n-th sublist L_n = {p_1, p_2, ..., p_m}. a(n) is the largest m such that the maximum prime gap in L_n is < p_1 - prevprime(p_1).

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A085500 Indices of primes where nondecreasing gaps occur.

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A361823 a(1) = 3; thereafter, a(n+1) is the smallest prime p such that p - prevprime(p) >= a(n) - prevprime(a(n)).

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