A335709 a(n) is the smallest Niven number that has exactly n divisors or -1 if no such number exists.

1, 2, 4, 6, 81, 12, -1, 24, 36, 48, 59049, 60, -1, 192, 144, 120, 43046721, 180, 43472473122830653562489222659449707872441, 240, 576, 3072, 191540580003116921429323712183642218614831262597249, 360, 1296, 94208, 900, 960, -1, 720, -1, 840, 9216, 720896, 5184, 1260

Offset: 1

Author

Marius A. Burtea, Aug 04 2020

Keywords

• sign
• base

Comments

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).

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.

Examples

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

Crossrefs

Cf. A000005, A005349, A007953.

Programs

Magma

niven:=func; 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;