A053303 -id:A053303 - cp's OEIS frontend

A038460 Maximal value of difference between successive primes among numbers < 10^n.

2, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, 1220, 1442, 1510

Offset: 1

Enoch Haga, Dec 11 1999

All terms are even, and the sequence is strictly increasing, and therefore also yields the maximal gap between n-digit primes (unless a gap containing 10^k would be larger than all gaps up to 10^(k+1), which does not happen). Therefore also a subsequence of A005250, which is a subsequence of A001223. - M. F. Hasler, Dec 29 2014

For 3 < n < 19, a(n) <= 6 (n - 1)(n - 2). Conjecture: for any n > 3, a(n) <= 6 (n - 1)(n - 2). Let q = 6 (n - 1)(n - 2) and d = (10^n) - (10^(n/2) - 1)^2. Since for any even n, d is the smallest difference between two consecutive squares of the form a^2 - b^2, where a^2 = 10^n, b = a - 1, for any even n > 2, d > 5q (where 3q is, according to the conjecture, not less than the sum of the three largest gaps between 4 consecutive primes p1...p4, or 3 * a(n), and 2q is, respectively, not less than the sum of the two largest gaps (p1 - p0) + (p5 - p4), or 2 * a(n)). In the same way, we can state that for any odd n > 3, if a^2 is the smallest square such that a^2 has (n+1) digits, b = a - 1, and d = a^2 - b^2, then d > 5q. The correctness of the above conjecture would establish the well-known Brocard's and Legendre's conjectures (see the link below for both definitions), since they are proved for the first 10000 primes. - Sergey Pavlov, Jan 30 2017

			Of the 25 primes less than 100, the maximum difference between two consecutive primes is 8 (at 97 - 89), so a(2)=8.

Enoch Haga, Exploring Prime Numbers on Your PC, 2nd edition, 1998, ISBN 1-885794-16-9, Table 3.

Jens Kruse Andersen, Record prime gaps, current version maintained by Norman Luhn.
C. K. Caldwell, Prime gaps
Wikipedia, Brocard's conjecture

Cf. A038343. See A053303 for another version.

a[1] = 2; a[n_] := a[n] = Module[{m, p, q}, m = a[n-1]; p = NextPrime[10^(n - 1), -1]; For[q = NextPrime[p], q < 10^n, q = NextPrime[q], m = Max[m, q - p]; p = q]; m]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 10}] (* Jean-François Alcover, May 16 2017 *)
Table[Differences[Prime[Range[PrimePi[10^n]]]]//Max,{n,10}] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Sep 11 2024 *)

a(n) = {dmax = 0; minp = 2; forprime(p=3, 10^n, if ((d = (p - minp)) > dmax, dmax = d); minp = p;); dmax;} \\ Michel Marcus, May 25 2014

m=0;L=1;o=2;forprime(p=3,10^9,m+o>=(o=p)&&next;#Str(p)>L&&!print1(m" /*"L"*/, ")&&L=#Str(p);m=p-precprime(p-1)) \\ Too slow for n>9. M. F. Hasler, Dec 29 2014

Two more terms (282, 354) from Jud McCranie
Terms 464 through 1442 from Manfred W. K. Huppertz (huppi-manni(AT)hesit.de), Aug 18 2009
a(19) from Hugo Pfoertner, using data from pzktupel webpage. Jan 21 2025

A053302 Largest n-digit prime at the start of a record in the RECORDS transform of the prime gaps.

Original entry on oeis.org

7, 89, 887, 9551, 31397, 492113, 4652353, 47326693, 436273009, 4302407359, 42652618343, 738832927927, 7177162611713, 90874329411493, 218209405436543, 1693182318746371

Offset: 1

Enoch Haga, Mar 05 2000

a(17) is probably 80873624627234849 and a(19) is probably 8822016561303449927. - Robert G. Wilson v, Mar 16 2004
a(18) is most probably 804212830686677669. - M. F. Hasler, Apr 25 2014
Is this a duplicate of A073861 (except for the first term)? - M. F. Hasler, Apr 25 2014

			a(5)=31397, the last 5-digit prime to begin a gap.

Thomas R. Nicely, Some Results of Computational Research in Prime Numbers [Local copy, pdf only] [ See local copy in A007053]
Eric Weisstein's World of Mathematics, Prime Gaps

The length of the gap is in A053303.

a(n) = { max p: p in A002386; p< 10^n}. - R. J. Mathar, Feb 01 2008

Corrected by Jud McCranie, Jan 03 2001
a(15) and a(16) from Robert G. Wilson v, Mar 16 2004
Edited by R. J. Mathar, Feb 01 2008

A123386 Largest difference between successive semiprimes up to 10^n inclusive.

Original entry on oeis.org

3, 7, 14, 24, 38, 47, 74, 74, 95, 112, 146, 163, 174

Offset: 1

Alexander Adamchuk, Nov 09 2006

There are 4 semiprimes up to 10^1 {4, 6, 9, 10}. The differences between successive semiprimes are {2, 3, 1}. Thus a(1) = Max[ {2, 3, 1} ] = 3.

Cf. A001358, A065516, A036352, A038460, A053303.

A001358(prev)={ local(a=prev+1) ; while(bigomega(a)!=2, a++ ; ) ; return(a) ; }
A123386(n)={ local(sp1=4,sp2=6,a=2) ; while(sp2<=10^n, a=max(a,sp2-sp1) ; sp1=sp2 ; sp2=A001358(sp1) ; ) ; return(a) ; }
{ for(n=1,13, print(A123386(n)) ; ) ; } \\ 2 more terms from R. J. Mathar, Jan 17 2008

2 more term from R. J. Mathar, Jan 17 2008
a(8)-a(9) from Donovan Johnson, Sep 05 2008
a(10)-a(11) from Donovan Johnson, Apr 14 2010
a(12)-a(13) from Donovan Johnson, Sep 20 2012

A286894 a(n) = maximum difference between two consecutive primes up to 2^n.

Original entry on oeis.org

0, 1, 2, 4, 6, 6, 14, 14, 14, 20, 34, 34, 34, 44, 72, 72, 72, 86, 114, 114, 148, 148, 154, 154, 210, 220, 222, 248, 282, 282, 292, 336, 354, 382, 456, 464, 468, 486, 532, 540, 602, 652, 674, 716, 766, 778, 804, 906, 906, 906

Offset: 1

Jean-François Alcover, May 15 2017

			Up to 2^5 = 32, the maximum difference between two consecutive primes is 6 (at 29 - 23), so a(5)=6.

Eric Weisstein's MathWorld, Prime Gaps

Cf. A038460, A053303.

a[n_] := If[n == 1, 0, Prime[Range[PrimePi[NextPrime[2^n, -1]]]] // Differences // Max]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 30}]

a(n) = {maxg = 0; p = 2; forprime(q=3, 2^n, maxg = max(maxg, q-p); p=q;); maxg;} \\ Michel Marcus, May 15 2017

a(33)-a(50) from David A. Corneth, May 16 2017

cp's OEIS Frontend

A038460 Maximal value of difference between successive primes among numbers < 10^n.

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A053302 Largest n-digit prime at the start of a record in the RECORDS transform of the prime gaps.

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A123386 Largest difference between successive semiprimes up to 10^n inclusive.

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PARI

Extensions

A286894 a(n) = maximum difference between two consecutive primes up to 2^n.

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Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions