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A038460 Maximal value of difference between successive primes among numbers < 10^n. 6
2, 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, 1220, 1442 (list; graph; refs; listen; history; text; internal format)
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
C. K. Caldwell, Prime gaps
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 *)
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
Cf. A038343. See A053303 for another version.
Sequence in context: A031114 A327100 A130238 * A077588 A025219 A278212
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

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Last modified February 22 15:38 EST 2024. Contains 370256 sequences. (Running on oeis4.)