A393306 - OEIS

%I #10 Feb 15 2026 22:47:37

%S 2,19,201,2902,44185,799899,16434817,381367036,9876541023,

%T 282458540465,8842413667417,300771807236682,11046255305795333,

%U 435659737878916201,18364758544492069665,824008854613343260616,39210261334551566850727,1972313422155189164329713,104567135734072022160160883

%N a(n) is the greatest number k that contains, in base n, every digit exactly once, and such that k/(n-1) (if n is even) or 2*k/(n-1) (if n is odd) is prime.

%C Every number that contains, in base n, every digit once is divisible by n-1 (if n is even) or (n-1)/2 (if n is odd).

%C Conjecture: a(n) exists for all n.

%H Robert Israel, <a href="/A393306/b393306.txt">Table of n, a(n) for n = 2..385</a>

%e a(4) = 201 because 201 = 3021_4 contains all digits 0 to 3 once in base 4, 201/(4-1) = 67 is prime, and no larger number works.

%p f:= proc(n) local b,p,x,i; uses combinat;

%p       b:= `if`(n::even,n-1,(n-1)/2);

%p       p:= lastperm(n);

%p       do

%p         x:= add((p[i]-1)*n^(n-i),i=1..n);

%p         if isprime(x/b) then return x fi;

%p         p:= prevperm(p);

%p         if p = FAIL then return -1 fi

%p       od;

%p end proc:

%p map(f, [$2..30]);

%Y Cf. A380386.

%K nonn,base

%O 2,1

%A _Robert Israel_, Feb 10 2026