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

1,2

Links

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

Formula

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

Example

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].

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 *)

Crossrefs

Cf. A122651, A184998.

Sequence in context: A123581 A187337 A371000 * A289761 A310151 A310152

Adjacent sequences: A184996 A184997 A184998 * A185000 A185001 A185002

Keyword

nonn,look

Author

Alois P. Heinz, Mar 28 2011

Status

approved