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A305566
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Number of finite sets of relatively prime positive integers > 1 with least common multiple n.
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12
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0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 10, 0, 2, 2, 0, 0, 10, 0, 10, 2, 2, 0, 44, 0, 2, 0, 10, 0, 84, 0, 0, 2, 2, 2, 122, 0, 2, 2, 44, 0, 84, 0, 10, 10, 2, 0, 184, 0, 10, 2, 10, 0, 44, 2, 44, 2, 2, 0, 1590, 0, 2, 10, 0, 2, 84, 0, 10, 2, 84, 0, 1156, 0, 2, 10, 10, 2
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OFFSET
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1,6
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COMMENTS
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a(n) depends only on the prime signature of n.
If n = p^k*q, where p and q are distinct primes and k >= 1, then a(n) = 3*4^(k-1)-2^(k-1). - Robert Israel, Jun 07 2018
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LINKS
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EXAMPLE
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The a(12) = 10 sets:
{3,4},
{2,3,4}, {2,3,12}, {3,4,6}, {3,4,12},
{2,3,4,6}, {2,3,4,12}, {2,3,6,12}, {3,4,6,12},
{2,3,4,6,12}.
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MAPLE
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f:= proc(n) g(sort(map(t -> t[2], ifactors(n)[2]))) end proc:
f(1):= 0:
g:= proc(L) option remember;
local nL, Cands, nC, Cons, i;
nL:= nops(L);
Cands:= [[]];
for i from 1 to nL do
Cands:= [seq(seq([op(s), t], t=0..L[i]), s=Cands)];
od:
Cands:= remove(t -> max(t)=0, Cands);
nC:= nops(Cands);
Cons:= [seq(select(t -> Cands[t][i]=0, {$1..nC}), i=1..nL),
seq(select(t -> Cands[t][i]=L[i], {$1..nC}), i=1..nL)];
h(Cons, {$1..nC})
end proc:
h:= proc(Cons, Cands)
local t, i, Consi, Candsi;
if Cons = [] then return 2^nops(Cands) fi;
t:= 0;
for i from 1 to nops(Cons[1]) do
Consi:= map(proc(t) if member(Cons[1][i], t) then NULL else t minus Cons[1][1..i-1] fi end proc, Cons[2..-1]);
if member({}, Consi) then next fi;
Candsi:= Cands minus Cons[1][1..i];
t:= t + procname(Consi, Candsi)
od;
t
end proc:
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MATHEMATICA
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Table[Length[Select[Subsets[Rest[Divisors[n]]], And[GCD@@#==1, LCM@@#==n]&]], {n, 100}]
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CROSSREFS
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Cf. A000961, A006881, A076078, A181819, A281116, A285572, A290103, A304818, A305563, A305564, A305565, A305567.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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