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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

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OFFSET

2,1

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

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

(SageMath) 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