A121382 Number of ways of writing n as x*y*z, with x <= y <= z and gcd(x,y) = gcd(y,z) = 1.

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

Offset

1,6

Comments

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).

Links

Robert Israel, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen)

Example

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.

Maple

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:
convert(A, list); # Robert Israel, Aug 27 2017

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A000040, A001248, A001358, A007304, A007875, A030078.

First occurrence of k: A122829.

Sequence in context: A333175 A294893 A336570 * A305150 A351202 A374959

Adjacent sequences: A121379 A121380 A121381 * A121383 A121384 A121385

Keyword

easy,nonn

Author

Jonathan Vos Post, Sep 06 2006

Extensions

Edited, corrected and extended by Robert G. Wilson v, Sep 11 2006

Name clarified by Antti Karttunen, Aug 27 2017

Status

approved