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

Author

Alois P. Heinz, Nov 14 2012

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

Maximal products are in A219209.

Cf. A033630, A092976.

Programs

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 *)