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A121382
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Number of ways of writing n as x*y*z, with x <= y <= z and gcd(x,y) = gcd(y,z) = 1.
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2
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1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 7, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 3, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 5
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OFFSET
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1,6
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COMMENTS
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3-factor analog of A007875 (number of ways of writing n as x*y, with x <= y and gcd(x,y)=1).
a(n) = 1 iff n is a prime power (A000961).
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LINKS
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EXAMPLE
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a(4) = 1 because 4 = 1*1*4.
a(6) = 2 because 6 = 1*1*6 = 1*2*3.
a(24) = 3 because 24 = 1*1*24 = 1*3*8 = 2*3*4.
a(30) = 5 because 30 = 1*1*30 = 1*2*15 = 1*3*10 = 1*5*6 = 2*3*5.
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MAPLE
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N:= 1000:
A:= Vector(N):
for y from 1 to floor(sqrt(N)) do
X:= select(t -> igcd(t, y)=1, [$1..y]);
Z:= select(t -> igcd(t, y)=1, [$y..N/y]);
for x in X do
for z in Z while x*y*z <= N do
A[x*y*z]:= A[x*y*z]+1
od od:
od:
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MATHEMATICA
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f[n_] := Block[{d = Divisors@n, m = DivisorSigma[0, n], s = {}}, If[m == 2, 1, Do[ AppendTo[s, {d[[p]], d[[q]], d[[r]]}], {r, m}, {q, r}, {p, q}]; Length@ Select[s, Times @@ # == n && GCD[ #[[1]], #[[2]]] == GCD[ #[[2]], #[[3]]] == 1 &]]]; Array[f, 105] (* Robert G. Wilson v, Sep 11 2006 *)
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PROG
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(PARI) A121382(n) = { my(s=0); fordiv(n, x, for(y=x, n, for(z=y, n, if((x*y*z==n)&&(1==gcd(x, y))&&(1==gcd(y, z)), s++)))); (s); }; \\ Antti Karttunen, Aug 27 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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