%I
%S 1,2,10,18,271
%N a(n) is the smallest number k > 0 such that for each b = 2..n the baseb expansion of k has exactly n  b zeros.
%C This list is complete. Proof: When converting base 2 to base 4, we can group the digits in base 2 into pairs from the least significant bit. We then convert pairs into single digits in base 4 as 00 > 0, 01 > 1, 10 > 2, 11 > 3. This always causes the number of zeros to go to half or less than half. For all n >= 7, n4 is greater than (n2)/2, so the condition is impossible.  _Christopher Cormier_, Dec 08 2019
%C Does k exist for every n >= 2?
%C a(7) > 10^7, if it exists.
%C a(7) > 2^64, if it exists.  _Giovanni Resta_, Dec 01 2019
%e For n = 6: The baseb expansions of 271 for b = 2..6 are shown in the following table:
%e b  baseb expansion  number of zeros
%e 
%e 2  100001111  4
%e 3  101001  3
%e 4  10033  2
%e 5  2041  1
%e 6  1131  0
%o (PARI) count_zeros(vec) = #setintersect(vecsort(vec), vector(#vec))
%o a(n) = for(k=1, oo, for(b=2, n, if(count_zeros(digits(k, b))!=nb, break, if(b==n, return(k)))))
%K nonn,base,fini,full
%O 2,2
%A _Felix FrÃ¶hlich_, Dec 01 2019
%E Value of a(2) adjusted by _Felix FrÃ¶hlich_, Dec 09 2019
