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

1, 2, 10, 36, 150, 180, 420, 840, 900, 3420, 2520, 5040, 6300, 7560, 12600, 15120, 18900, 42840, 32760, 37800, 95760, 105840, 69300, 124740, 163800, 138600, 166320, 327600, 249480, 207900, 491400, 622440, 498960, 706860, 415800, 963900, 1496880, 1164240, 1081080

Offset: 0

Author

Marius A. Burtea, Sep 20 2022

Keywords

• nonn
• base

Examples

			1 has no divisors in A031443, so a(0) = 1;
2 has divisors 1 = 1_2, 2 = 10_2 and 2 = A031443(1), so a(1) = 2.
10 has divisors 2 = 10_2 and 10 = 1010_2 in A031443, so a(2) = 10.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..97

Crossrefs

Cf. A031443.

Programs

Magma

bal:=func; 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;

Maple

N:= 20: # for terms before the first term >= 2^(N+1)
W:= Vector(2^(N+1),datatype=integer[4]):
for d from 2 to N by 2 do
for t from 2^(d-1) to 2^d-1 do
  if convert(convert(t,base,2),`+`) = d/2 then
    J:= [seq(i,i=t..2^(N+1), t)];
    W[J]:= W[J] +~ 1;
fi od od:
M:= max(W);
V:= Array(0..M); count:= 0:
for i from 1 to 2^(N+1) do
  if V[W[i]] = 0 then V[W[i]]:= i; count:= count+1 fi
od:
L:= convert(V,list):
if not member(0,L,'m') then m:= M+2 fi:
L[1..m-1]; # Robert Israel, Sep 27 2023

Mathematica

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 *)

Extensions

Corrected by Robert Israel, Sep 27 2023