A219208 Number of distinct products of all parts of all partitions of n into distinct divisors of n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 1, 1, 1, 1, 7, 1, 1, 1, 4, 1, 3, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 26, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 2, 1, 6, 1, 1, 1, 23, 1

Offset

0,13

Comments

a(p) = 1 for p in A000040 (prime numbers).

a(n) = 1 for n in A006037 (weird numbers).

a(n) = 1 for n in A005100 (deficient numbers).

a(n) = 1 for n in A125493 (composite deficient numbers).

a(n) <= 2 for n in A000396 (perfect numbers).

a(n) >= 2 for n > 6 and n in A005835 (semiperfect numbers).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Example

a(0) = 1: the empty product.
a(p) = 1 for any prime p: [p]-> p.
a(6) = 1: {[1,2,3], [6]}-> 6.
a(12) = 3, because all 3 partitions of 12 into distinct divisors of 12 have different products: [1,2,3,6]-> 36, [2,4,6]-> 48, [12]-> 12. a(18) = 3: [1,2,6,9]-> 108, [3,6,9]-> 162, [18]-> 18.
a(20) = 2: [1,4,5,10]-> 200, [20]-> 20.
a(28) = 2: [1,2,4,7,14]-> 784, [28]-> 28.
a(36) = 7: [2,3,4,6,9,12]-> 15552, [2,3,4,9,18]-> 3888, [1,2,6,9,18]-> 1944, [3,6,9,18]-> 2916, {[1,2,3,12,18], [6,12,18]}-> 1296, [2,4,12,18]-> 1728, [36]-> 36.
a(84) = 23: 84, 16464, 28224, 49392, 65856, 74088, 84672, 86436, 98784, 127008, 148176, 190512, 254016, 444528, 592704, 889056, 1016064, 1185408, 1382976, 1778112, 2370816, 4148928, 7112448.

Maple

a:= proc(n) local b, l;
      l:= sort([numtheory[divisors](n)[]]);
      b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},
            {b(n, i-1)[], `if`(l[i]>n, {}, map(x-> x*l[i],
            b(n-l[i], i-1)))[]}))
          end; forget(b);
      nops(b(n, nops(l)))
    end:
seq(a(n), n=0..120);

Mathematica

a[n_] := a[n] = Module[{b, l}, l = Divisors[n]; b[m_, i_] := b[m, i] = If[m == 0, {1}, If[i<1, {}, Union[b[m, i-1], If[l[[i]]>m, {}, (#*l[[i]]&) /@ b[m-l[[i]], i-1]]]]]; Length[b[n, Length[l]]]]; Table[a[n], {n, 0, 120}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)

Crossrefs

Maximal products are in A219209.

Cf. A033630, A092976.

Sequence in context: A180173 A318668 A322361 * A336850 A061680 A355456

Adjacent sequences: A219205 A219206 A219207 * A219209 A219210 A219211

Keyword

nonn

Author

Alois P. Heinz, Nov 14 2012

Status

approved