A333456 a(n) is the smallest number with exactly n divisors that are Niven numbers.

1, 2, 4, 6, 45, 12, 30, 24, 36, 84, 60, 72, 400, 144, 120, 216, 180, 240, 420, 504, 600, 960, 360, 900, 840, 1200, 1512, 720, 1620, 4500, 1080, 2700, 1800, 3024, 3360, 2880, 3960, 2160, 3240, 5760, 2520, 6720, 4320, 5400, 9360, 7920, 7200, 6480, 13860, 8640

Offset: 1

Author

Marius A. Burtea, May 03 2020

Keywords

• nonn
• base

Comments

A Niven number (A005349) is a number that is divisible by the sum of its digits; e.g., 12 is a Niven number because it's divisible by 1 + 2.

The divisor 1 is trivially a Niven number. For a number n, the divisor n itself is considered

Conjecture: For every n there is at least one number k with n divisors Niven numbers.

Not all terms in the sequence are Niven numbers. For example: a(85) = 85680 has digsum(85680) = 27 and 85680/27 = 3173.33 ...

Also, a(152) = 856800, a(159) = 887040, a(161) = 2096640.

The number of non-Niven terms can be infinite.

Examples

			Of the divisors of 45, only five are Niven numbers: 1, 3, 5, 9, and 45.
Number 12 has all six divisors 1, 2, 3, 4, 6 and 12 that are Niven numbers.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..297
• David A. Corneth, Some upper bounds

Crossrefs

Cf. A005349, A332268.

Programs

Magma

a:=[]; for n in [1..50] do m:=1; while #[d:d in Divisors(m)|d mod &+Intseq(d) eq 0] ne n do m:=m+1; end while; Append(~a,m); end for; a;

Mathematica

numDiv[n_] := DivisorSum[n, 1 &, Divisible[#, Plus @@ IntegerDigits[#]] &]; a[n_] := Module[{k = 1}, While[numDiv[k] != n, k++]; k]; Array[a, 50] (* Amiram Eldar, May 04 2020 *)

PARI

f(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ A332268
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, May 04 2020