A070866 -id:A070866 - 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

A070865 Smallest prime such that the difference of successive terms is strictly increasing.

Original entry on oeis.org

2, 3, 5, 11, 19, 29, 41, 59, 79, 101, 127, 157, 191, 227, 269, 313, 359, 409, 461, 521, 587, 659, 733, 809, 887, 967, 1049, 1151, 1259, 1373, 1489, 1607, 1733, 1861, 1993, 2129, 2267, 2411, 2557, 2707, 2861, 3019, 3181, 3347, 3517, 3691, 3877, 4073, 4271

Offset: 1

Amarnath Murthy, May 16 2002

nonn

The slowest increasing sequence of primes with strictly increasing difference of successive terms. - Zak Seidov and Charles R Greathouse IV, May 01 2015

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A070866.

d=0; p=2; t={p}; Do[d=NextPrime[p+d]-p; AppendTo[t,p+=d], {99}]; t (* Vladimir Joseph Stephan Orlovsky, May 29 2010 *)
nxt[{a_,b_}]:={b,NextPrime[2b-a]}; Transpose[NestList[nxt,{2,3},50]][[1]] (* Harvey P. Dale, Jan 04 2015 *)

t=0; print1(last=2); while(1, n=last+t; while(!isprime(n++), ); print1(", "n); t=n-last; last=n) \\ Charles R Greathouse IV, Apr 30 2015

a(n) >> n^2. - Charles R Greathouse IV, Apr 30 2015

Corrected and extended by Lior Manor, Jun 02 2002

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

A178549 a(n) is a composite number at the start of an interval of consecutive integers, ending in a prime, and non-overlapping with and at least as long as the interval addressed by a(n-1).

Original entry on oeis.org

4, 6, 8, 12, 18, 24, 30, 38, 48, 60, 72, 84, 98, 114, 132, 150, 168, 192, 224, 258, 294, 332, 374, 420, 468, 522, 578, 642, 710, 788, 878, 968, 1062, 1164, 1278, 1400, 1524, 1658, 1802, 1950, 2100, 2252, 2412, 2580, 2750, 2928, 3110, 3300, 3492, 3692, 3908

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 29 2010

nonn

Non-overlapping intervals of the integers of nondecreasing length, starting with a composite number and ending with a prime number, define an intermediate table (as below), the first column of which defines the sequence.
   4,  5
   6,  7
   8,  9, 10, 11
  12, 13, 14, 15, 16, 17
  18, 19, 20, 21, 22, 23
  24, 25, 26, 27, 28, 29
  30, 31, 32, 33, 34, 35, 36, 37
  38, 39, 40, 41, 42, 43, 44, 45, 46, 47
  48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59
  60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Cf. A070865, A178547.

PrimeNext[n_]:=Module[{k},k=n+1;While[ !PrimeQ[k],k++ ];k]; d=0;p=2;lst={}; Do[d=PrimeNext[p+d]-p;p=p+=d;d--;AppendTo[lst,p+1],{n,0,5!}];lst

a(n) = 1 + A070866(n+1). - R. J. Mathar, Jun 07 2010

Definition rephrased by R. J. Mathar, Jun 07 2010

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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A070865 Smallest prime such that the difference of successive terms is strictly increasing.

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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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A178549 a(n) is a composite number at the start of an interval of consecutive integers, ending in a prime, and non-overlapping with and at least as long as the interval addressed by a(n-1).

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