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
Keywords
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
Programs
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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 *)
Comments