A184999 Smallest number having exactly n partitions into distinct parts, with each part divisible by the next.

0, 3, 6, 9, 12, 15, 22, 25, 21, 30, 48, 36, 40, 56, 51, 45, 57, 64, 84, 76, 63, 90, 85, 93, 81, 99, 100, 91, 150, 130, 105, 133, 126, 147, 154, 184, 135, 153, 198, 213, 175, 304, 165, 265, 232, 183, 320, 171, 226, 210, 201, 274, 300, 243

Offset: 1

Author

Alois P. Heinz, Mar 28 2011

Keywords

• nonn
• look

Examples

			a(7) = 22, because A122651(22) = 7 and A122651(m) <> 7 for all m<22.  The 7 partitions of 22 into distinct parts, with each part divisible by the next are: [22], [21,1], [20,2], [18,3,1], [16,4,2], [14,7,1], [12,6,3,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A122651, A184998.

Programs

Maple

with(numtheory):
a:= proc() local t, a, b, bb;
      t:= -1;
      a:= proc() -1 end;
      bb:= proc(n) option remember;
        `if`(n=0, 1, add(bb((n-d)/d), d=divisors(n) minus{1}))
      end:
      b:= n-> `if`(n=0, 1, bb(n)+bb(n-1));
      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=1..100);

Mathematica

b[0]=1; b[n_] := b[n] = Sum[b[(n-d)/d], {d, Divisors[n] // Rest}]; a[0] = 1; a[n_] := For[k=0, True, k++, If[b[k]+b[k-1] == n, Return[k]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 03 2014, after Alois P. Heinz *)

Formula

a(n) = min { k : A122651(k) = n }.