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

%I A333456 #29 Sep 08 2022 08:46:25
%S A333456 1,2,4,6,45,12,30,24,36,84,60,72,400,144,120,216,180,240,420,504,600,
%T A333456 960,360,900,840,1200,1512,720,1620,4500,1080,2700,1800,3024,3360,
%U A333456 2880,3960,2160,3240,5760,2520,6720,4320,5400,9360,7920,7200,6480,13860,8640
%N A333456 a(n) is the smallest number with exactly n divisors that are Niven numbers.
%C A333456 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.
%C A333456 The divisor 1 is trivially a Niven number. For a number n, the divisor n itself is considered
%C A333456 Conjecture: For every n there is at least one number k with n divisors Niven numbers.
%C A333456 Not all terms in the sequence are Niven numbers. For example: a(85) = 85680 has digsum(85680) = 27 and 85680/27 = 3173.33 ...
%C A333456 Also, a(152) = 856800, a(159) = 887040, a(161) = 2096640.
%C A333456 The number of non-Niven terms can be infinite.
%H A333456 David A. Corneth, <a href="/A333456/b333456.txt">Table of n, a(n) for n = 1..297</a>
%H A333456 David A. Corneth, <a href="/A333456/a333456.gp.txt">Some upper bounds</a>
%e A333456 Of the divisors of 45, only five are Niven numbers: 1, 3, 5, 9, and 45.
%e A333456 Number 12 has all six divisors 1, 2, 3, 4, 6 and 12 that are Niven numbers.
%t A333456 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 *)
%o A333456 (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;
%o A333456 (PARI) f(n) = sumdiv(n, d, !(d % sumdigits(d))); \\ A332268
%o A333456 a(n) = my(k=1); while (f(k) != n, k++); k; \\ _Michel Marcus_, May 04 2020
%Y A333456 Cf. A005349, A332268.
%K A333456 nonn,base
%O A333456 1,2
%A A333456 _Marius A. Burtea_, May 03 2020