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

%I A184999 #25 Dec 06 2014 21:48:15
%S A184999 0,3,6,9,12,15,22,25,21,30,48,36,40,56,51,45,57,64,84,76,63,90,85,93,
%T A184999 81,99,100,91,150,130,105,133,126,147,154,184,135,153,198,213,175,304,
%U A184999 165,265,232,183,320,171,226,210,201,274,300,243
%N A184999 Smallest number having exactly n partitions into distinct parts, with each part divisible by the next.
%H A184999 Alois P. Heinz, <a href="/A184999/b184999.txt">Table of n, a(n) for n = 1..10000</a>
%F A184999 a(n) = min { k : A122651(k) = n }.
%e A184999 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].
%p A184999 with(numtheory):
%p A184999 a:= proc() local t, a, b, bb;
%p A184999       t:= -1;
%p A184999       a:= proc() -1 end;
%p A184999       bb:= proc(n) option remember;
%p A184999         `if`(n=0, 1, add(bb((n-d)/d), d=divisors(n) minus{1}))
%p A184999       end:
%p A184999       b:= n-> `if`(n=0, 1, bb(n)+bb(n-1));
%p A184999       proc(n) local h;
%p A184999         while a(n) = -1 do
%p A184999           t:= t+1;
%p A184999           h:= b(t);
%p A184999           if a(h) = -1 then a(h):= t fi
%p A184999         od; a(n)
%p A184999       end
%p A184999     end():
%p A184999 seq(a(n), n=1..100);
%t A184999 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_ *)
%Y A184999 Cf. A122651, A184998.
%K A184999 nonn,look
%O A184999 1,2
%A A184999 _Alois P. Heinz_, Mar 28 2011