

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
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OFFSET

1,1


COMMENTS

All terms are even, and the sequence is strictly increasing, and therefore also yields the maximal gap between ndigit 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 wellknown 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 1885794169, 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[n1]; 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}] (* JeanFrançois 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=pprecprime(p1)) \\ Too slow for n>9. M. F. Hasler, Dec 29 2014


CROSSREFS



KEYWORD

nonn,more,nice


AUTHOR



EXTENSIONS

Terms 464 through 1442 from Manfred W. K. Huppertz (huppimanni(AT)hesit.de), Aug 18 2009


STATUS

approved



