OFFSET
1,1
COMMENTS
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
REFERENCES
Enoch Haga, Exploring Prime Numbers on Your PC, 2nd edition, 1998, ISBN 1-885794-16-9, Table 3.
LINKS
EXAMPLE
Of the 25 primes less than 100, the maximum difference between two consecutive primes is 8 (at 97 - 89), so a(2)=8.
MATHEMATICA
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 *)
PROG
(PARI) 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
(PARI) 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
CROSSREFS
KEYWORD
nonn,more,nice
AUTHOR
Enoch Haga, Dec 11 1999
EXTENSIONS
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
STATUS
approved