A219208 - OEIS

%I #36 Feb 16 2017 11:01:00

%S 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,

%T 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,

%U 1,1,1,1,1,26,1,1,1,1,1,2,1,6,1,1,1,23,1

%N Number of distinct products of all parts of all partitions of n into distinct divisors of n.

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

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

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

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

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

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

%H Alois P. Heinz, <a href="/A219208/b219208.txt">Table of n, a(n) for n = 0..5000</a>

%e a(0) = 1: the empty product.

%e a(p) = 1 for any prime p: [p]-> p.

%e a(6) = 1: {[1,2,3], [6]}-> 6.

%e 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.

%e a(20) = 2: [1,4,5,10]-> 200, [20]-> 20.

%e a(28) = 2: [1,2,4,7,14]-> 784, [28]-> 28.

%e 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.

%e 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.

%p a:= proc(n) local b, l;

%p       l:= sort([numtheory[divisors](n)[]]);

%p       b:= proc(n, i) option remember; `if`(n=0, {1}, `if`(i<1, {},

%p             {b(n, i-1)[], `if`(l[i]>n, {}, map(x-> x*l[i],

%p             b(n-l[i], i-1)))[]}))

%p           end; forget(b);

%p       nops(b(n, nops(l)))

%p     end:

%p seq(a(n), n=0..120);

%t 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 *)

%Y Maximal products are in A219209.

%Y Cf. A033630, A092976.

%K nonn

%O 0,13

%A _Alois P. Heinz_, Nov 14 2012