%I #18 Sep 08 2022 08:46:25

%S 1,2,4,6,81,12,-1,24,36,48,59049,60,-1,192,144,120,43046721,180,

%T 43472473122830653562489222659449707872441,240,576,3072,

%U 191540580003116921429323712183642218614831262597249,360,1296,94208,900,960,-1,720,-1,840,9216,720896,5184,1260

%N a(n) is the smallest Niven number that has exactly n divisors or -1 if no such number exists.

%C If n is a prime number, then a(n) has the form p^(n-1), where p is a prime number such that p <= 9 * ((n-1) * log_10(p) +1).

%C For p <= 9 * ((n-1) * log_10 (p) +1), if there is no s >= 1 such that digsum(p^(n-1)) = p^s, then a(n) = -1. For example, for n = 7, the largest prime number p verifying p <= 9 * (6 * log_10 (p) +1) is 113, but no prime number q <= 113 has the property digsum(q^6) = q^s, for 1 <= s <= 6. Thus, a(7) = -1.

%e The number 81 = 3^4 is the smallest with 5 divisors and is a Niven number, so a(5) = 81.

%o (Magma) niven:=func<n|n mod &+Intseq(n) eq 0 >; a:=[]; for n in [1..36] do if not IsPrime(n) then k:=1; while not niven(k) or #Divisors(k) ne n do k:=k+1; end while; Append(~a,k); else q:=2; while not niven(q^(n-1)) and q le (9*(n-1)*Log(10,q)+9) do q:=NextPrime(q); end while; if niven(q^(n-1)) then Append(~a,q^(n-1)); else Append(~a,-1); end if; end if; end for; a;

%Y Cf. A000005, A005349, A007953.

%K sign,base

%O 1,2

%A _Marius A. Burtea_, Aug 04 2020