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A357035 a(n) is the smallest number that has exactly n divisors that are digitally balanced numbers (A031443).

%I A357035 #17 Sep 27 2023 11:23:05
%S A357035 1,2,10,36,150,180,420,840,900,3420,2520,5040,6300,7560,12600,15120,
%T A357035 18900,42840,32760,37800,95760,105840,69300,124740,163800,138600,
%U A357035 166320,327600,249480,207900,491400,622440,498960,706860,415800,963900,1496880,1164240,1081080
%N A357035 a(n) is the smallest number that has exactly n divisors that are digitally balanced numbers (A031443).
%H A357035 Robert Israel, <a href="/A357035/b357035.txt">Table of n, a(n) for n = 0..97</a>
%e A357035 1 has no divisors in A031443, so a(0) = 1;
%e A357035 2 has divisors 1 = 1_2, 2 = 10_2 and 2 = A031443(1), so a(1) = 2.
%e A357035 10 has divisors 2 = 10_2 and 10 = 1010_2 in A031443, so a(2) = 10.
%p A357035 N:= 20: # for terms before the first term >= 2^(N+1)
%p A357035 W:= Vector(2^(N+1),datatype=integer[4]):
%p A357035 for d from 2 to N by 2 do
%p A357035 for t from 2^(d-1) to 2^d-1 do
%p A357035   if convert(convert(t,base,2),`+`) = d/2 then
%p A357035     J:= [seq(i,i=t..2^(N+1), t)];
%p A357035     W[J]:= W[J] +~ 1;
%p A357035 fi od od:
%p A357035 M:= max(W);
%p A357035 V:= Array(0..M); count:= 0:
%p A357035 for i from 1 to 2^(N+1) do
%p A357035   if V[W[i]] = 0 then V[W[i]]:= i; count:= count+1 fi
%p A357035 od:
%p A357035 L:= convert(V,list):
%p A357035 if not member(0,L,'m') then m:= M+2 fi:
%p A357035 L[1..m-1]; # _Robert Israel_, Sep 27 2023
%t A357035 digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ @ (m = Length[d]) && Count[d, 1] == m/2]; f[n_] := DivisorSum[n, 1 &, digBalQ[#] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[40, 10^7] (* _Amiram Eldar_, Sep 26 2022 *)
%o A357035 (Magma) bal:=func<n|Multiplicity(Intseq(n,2),1) eq Multiplicity(Intseq(n,2),0)>; a:=[]; for n in [0..38] do k:=1; while #[d:d in Divisors(k)|bal(d)] ne n  do k:=k+1; end while; Append(~a,k); end for; a;
%Y A357035 Cf. A031443.
%K A357035 nonn,base
%O A357035 0,2
%A A357035 _Marius A. Burtea_, Sep 20 2022
%E A357035 Corrected by _Robert Israel_, Sep 27 2023