A132129 Largest prime with distinct digits when written in base n.

2, 19, 19, 577, 7417, 114229, 2053313, 42373937, 987654103, 25678048763, 736867805209, 23136292864193, 789018236128391, 29043982525257901, 1147797409030815779, 48471109094902530293, 2178347851919531491093, 103805969587115219167613, 5228356786703601108008083

Offset

2,1

Comments

a(10) = 987654103 = A007810(9). For n >= 3, a(n) < A062813(n), a multiple of n.

Contribution R. J. Mathar, May 15 2010 (START):

Supposed all digits are used and the digits at positions 0 to n-1 are d_0, d_1,... d_{n-1}, the candidates are d_0+d_1*n+d_2*n^2+....+d_{n-1}*n^(n-1).

These values are (n-1)*n/2 (mod n-1), and they cannot be prime if n is even, because this number is = 0 (mod n-1) then, showing that n-1 is a divisor.

In conclusion, if n is even, the entries have at most n-1 digits in base n. (END)

If n is odd then the candidate numbers considered in the previous comment are divisible by (n-1)/2. Hence, we conclude that for n>3, a(n) has at most n-1 digits in base n. Conjecture: for n>3, a(n) has exactly n-1 digits in base n. - Eric M. Schmidt, Oct 26 2014

Links

Eric M. Schmidt, Table of n, a(n) for n = 2..200

Example

a(9) = 42373937 as the prime 42373937 (base 10) = 87654102 (base 9), the largest prime number with distinct digits when represented in base 9.

Prog

(Sage) def a(n) :
    if n==2 : return 2
    if n==3 : return 19
    for P in Permutations(range(n-1, -1, -1), n-1) :
        N = sum(P[-1-i]*n^i for i in range(n-1))
        if is_prime(N) : return N
# Eric M. Schmidt, Oct 26 2014

Crossrefs

Cf. A062813, A007810, A029743.

Sequence in context: A065643 A038031 A229264 * A125611 A340922 A370460

Adjacent sequences: A132126 A132127 A132128 * A132130 A132131 A132132

Keyword

base,nonn

Author

Rick L. Shepherd, Aug 11 2007

Extensions

Removed my claim of finiteness of the sequence. - R. J. Mathar, May 18 2010

a(11)-a(20) from Eric M. Schmidt, Oct 26 2014

Status

approved