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

Author

Rick L. Shepherd, Aug 11 2007

Keywords

• base
• nonn

Comments

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

Contribution from 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

Examples

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

Links

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

Crossrefs

Cf. A062813, A007810, A029743.

Programs

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

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