A184999 -id:A184999 - cp's OEIS frontend

A122651 Number of partitions of n into distinct parts, with each part divisible by the next.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 3, 5, 5, 4, 6, 6, 4, 6, 6, 6, 9, 7, 4, 7, 8, 7, 9, 9, 6, 10, 10, 7, 10, 8, 8, 12, 9, 7, 12, 13, 8, 12, 12, 9, 16, 12, 5, 11, 13, 13, 15, 13, 9, 12, 15, 14, 17, 13, 7, 14, 14, 11, 21, 18, 13, 21, 16, 10, 14, 16, 12, 15, 15, 10, 21, 20, 13, 20, 16, 17, 25, 17, 9, 19

Offset: 0

Zak Seidov, Franklin T. Adams-Watters and Vladeta Jovovic, Sep 21 2006

			a(9)  = 4 : [9], [8,1], [6,3], [6,2,1].
a(15) = 6 : [15], [14,1], [12,3], [12,2,1], [10,5], [8,4,2,1].

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

Cf. A003238, A122934, A167439, A167865, A167866, A184999.

A122651r := proc(n,pmax,dv) option remember ; local a,d ; a := 0 ; for d in dv do if d = n and d <= pmax then a := a+1 ; elif d < pmax and n-d > 0 then a := a+A122651r(n-d,d-1,numtheory[divisors](d) minus {d} ) ; fi; od: a ; end: A122651 := proc(n) local i; A122651r(n,n, convert([seq(i,i=1..n)],set) ) ; end: for n from 1 to 120 do printf("%d,",A122651(n)) ; od:  # R. J. Mathar, May 22 2009
# second Maple program:
with(numtheory):
b:= proc(n) option remember;
      `if`(n=0, 1, add(b((n-d)/d), d=divisors(n) minus{1}))
    end:
a:= n-> `if`(n=0, 1, b(n)+b(n-1));
seq(a(n), n=0..200);  # Alois P. Heinz, Mar 28 2011

b[0] = 1; b[n_] := b[n] = Sum[b[(n - d)/d], {d, Divisors[n] // Rest}]; a[0] = 1; a[n_] := b[n] + b[n-1]; Table[a[n], {n, 0, 84}] (* Jean-François Alcover, Mar 26 2013, after Alois P. Heinz *)

{ a(n,m=0) = local(r=0); if(n==0,return(1)); fordiv(n,d, if(d<=m,next); r+=a((n-d)\d,1); ); r } /* Max Alekseyev */

For n>0, a(n) = A167865(n) + A167865(n-1).

More terms from R. J. Mathar, May 22 2009

a(0)=1 prepended by Max Alekseyev, Nov 13 2009

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

Original entry on oeis.org

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

Alois P. Heinz, Mar 28 2011

			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].

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

Cf. A167865, A184999.

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);

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 *)

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

cp's OEIS Frontend

A122651 Number of partitions of n into distinct parts, with each part divisible by the next.

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