cp's OEIS Frontend

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

%I A132129 #16 Aug 15 2025 00:58:45
%S A132129 2,19,19,577,7417,114229,2053313,42373937,987654103,25678048763,
%T A132129 736867805209,23136292864193,789018236128391,29043982525257901,
%U A132129 1147797409030815779,48471109094902530293,2178347851919531491093,103805969587115219167613,5228356786703601108008083
%N A132129 Largest prime with distinct digits when written in base n.
%C A132129 a(10) = 987654103 = A007810(9). For n >= 3, a(n) < A062813(n), a multiple of n.
%C A132129 Contribution from _R. J. Mathar_, May 15 2010: (Start)
%C A132129 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).
%C A132129 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.
%C A132129 In conclusion, if n is even, the entries have at most n-1 digits in base n. (End)
%C A132129 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
%H A132129 Eric M. Schmidt, <a href="/A132129/b132129.txt">Table of n, a(n) for n = 2..200</a>
%e A132129 a(9) = 42373937 as the prime 42373937 (base 10) = 87654102 (base 9), the largest prime number with distinct digits when represented in base 9.
%o A132129 (Sage) def a(n) :
%o A132129     if n==2 : return 2
%o A132129     if n==3 : return 19
%o A132129     for P in Permutations(range(n-1,-1,-1), n-1) :
%o A132129         N = sum(P[-1-i]*n^i for i in range(n-1))
%o A132129         if is_prime(N) : return N
%o A132129 # _Eric M. Schmidt_, Oct 26 2014
%Y A132129 Cf. A062813, A007810, A029743.
%K A132129 base,nonn
%O A132129 2,1
%A A132129 _Rick L. Shepherd_, Aug 11 2007
%E A132129 Removed my claim of finiteness of the sequence. - _R. J. Mathar_, May 18 2010
%E A132129 a(11)-a(20) from _Eric M. Schmidt_, Oct 26 2014