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URL: https://oeis.org/A333456

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A333456
a(n) is the smallest number with exactly n divisors that are Niven numbers.
14
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
1,2
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.
LINKS
Michael S. Branicky, Table of n, a(n) for n = 1..1214 (terms 1..297 from David A. Corneth)
David A. Corneth, Some upper bounds
EXAMPLE
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.
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 *)
PROG
(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;
(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
CROSSREFS
Sequence in context: A344422 A084324 A329529 * A377700 A348244 A381085
KEYWORD
nonn,base
AUTHOR
Marius A. Burtea, May 03 2020
STATUS
approved