A184998 Smallest number having exactly n partitions into distinct parts greater than 1, with each part divisible by the next.

1, 0, 6, 14, 12, 18, 24, 40, 36, 30, 48, 42, 75, 60, 72, 66, 80, 105, 84, 114, 102, 90, 120, 138, 132, 126, 186, 156, 150, 170, 180, 182, 310, 222, 200, 272, 434, 234, 198, 320, 273, 308, 210, 354, 252, 300, 360, 372, 392, 500, 366, 315

Offset: 0

Author

Alois P. Heinz, Mar 28 2011

Keywords

• nonn
• look

Examples

			a(7) = 40, because A167865(40) = 7 and A167865(m) <> 7 for all m<40.  The 7 partitions of 40 into distinct parts greater than 1, with each part divisible by the next are: [40], [38,2], [36,4], [35,5], [32,8], [30,10], [24,12,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A167865, A184999.

Programs

Maple

with(numtheory):
a:= proc() local t, a, b;
      t:= -1;
      a:= proc() -1 end;
      b:= proc(n) option remember;
            `if`(n=0, 1, add(b((n-d)/d), d=divisors(n) minus{1}))
          end:
      proc(n) local h;
        while a(n) = -1 do
          t:= t+1;
          h:= b(t);
          if a(h) = -1 then a(h):= t fi
        od; a(n)
      end
    end():
seq(a(n), n=0..100);

Mathematica

a[n0_] := Module[{t = -1, a, b}, a[] = -1; b[n] := b[n] = If[n == 0, 1, Sum[b[(n - d)/d], {d, Divisors[n] ~Complement~ {1}}]]; While[a[n] == -1, t++; h = b[t]; If[a[h] == -1, a[h] = t]]; a[n0]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, May 21 2018, translated from Maple *)

Formula

a(n) = min { k : A167865(k) = n }.