A086222 -id:A086222 - cp's OEIS frontend

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.

Original entry on oeis.org

1, 7, 7, 19, 7, 49, 7, 37, 19, 49, 7, 133, 7, 49, 49, 61, 7, 133, 7, 133, 49, 49, 7, 259, 19, 49, 37, 133, 7, 343, 7, 91, 49, 49, 49, 361, 7, 49, 49, 259, 7, 343, 7, 133, 133, 49, 7, 427, 19, 133, 49, 133, 7, 259, 49, 259, 49, 49, 7, 931, 7, 49, 133, 127, 49, 343, 7, 133

Offset: 1

Benoit Cloitre, May 20 2002

A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.

Antti Karttunen, Table of n, a(n) for n = 1..10000
O. Bagdasar, On some functions involving the lcm and gcd of integer tuples, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.

Cf. A000005, A008683, A048691, A070920, A070921, A086222, A247513.

Join[{1},Table[Product[(k + 1)^3 - k^3, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* Geoffrey Critzer, Jan 10 2015 *)

for(n=1,100,print1(sumdiv(n,d,numdiv(d)^3*moebius(n/d)),","))

a(n) = vecprod(apply(x->(x+1)^3-x^3, factor(n)[, 2])); \\ Amiram Eldar, Sep 03 2023

a(n) = Sum_{d|n} A000005(d)^3*A008683(n/d).
Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^3/k^s.
Multiplicative with a(p^e) = 1+3*e+3*e^2 for prime p and e >= 0. - Werner Schulte, Nov 30 2018

A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

Original entry on oeis.org

0, 0, 0, 1, 0, 4, 0, 3, 1, 4, 0, 15, 0, 4, 4, 6, 0, 15, 0, 15, 4, 4, 0, 33, 1, 4, 3, 15, 0, 44, 0, 10, 4, 4, 4, 48, 0, 4, 4, 33, 0, 44, 0, 15, 15, 4, 0, 58, 1, 15, 4, 15, 0, 33, 4, 33, 4, 4, 0, 133, 0, 4, 15, 15, 4, 44, 0, 15, 4, 44, 0, 100, 0, 4, 15, 15, 4, 44, 0, 58, 6, 4, 0, 133, 4, 4, 4, 33, 0

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 13 2003

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A048691, A070919, A086222.

for n from 1 to 100 do a[n] := 0:for x from 1 to n do for y from x+1 to n do for z from y+1 to n do if(lcm(x,y,z)=n) then a[n] := a[n]+1:fi:od:od:od:od:seq(a[j],j=1..200); # Sascha Kurz, Sep 22 2003

f1[p_, e_] := (e+1)^3 - e^3; f2[p_, e_] := 2*e + 1; a[1] = 0; a[n_] := (Times @@ f1 @@@ (f = FactorInteger[n]) - 3 * Times @@ f2 @@@f + 2) / 6; Array[a, 100] (* Amiram Eldar, Sep 03 2023 *)

A048691(n) = numdiv(n^2);
A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));
A086165(n) = ((A070919(n)-3*A048691(n)+2)/6); \\ Antti Karttunen, May 19 2017, after Jovovic's formula

a(n) = {my(e = factor(n)[, 2]); (vecprod(apply(x->(x+1)^3-x^3, e)) - 3*vecprod(apply(x->2*x+1, e)) + 2) / 6;} \\ Amiram Eldar, Sep 03 2023

a(n) = (A070919(n) - 3*A048691(n) + 2)/6. - Vladeta Jovovic, Dec 01 2004

a(n) = A086222(n) - A048691(n). - Ridouane Oudra, Aug 14 2025

More terms from Sascha Kurz, Sep 22 2003

A377304 a(n) is the number of distinct cuboids whose edges are divisors of n.

Original entry on oeis.org

1, 4, 4, 10, 4, 20, 4, 20, 10, 20, 4, 56, 4, 20, 20, 35, 4, 56, 4, 56, 20, 20, 4, 120, 10, 20, 20, 56, 4, 120, 4, 56, 20, 20, 20, 165, 4, 20, 20, 120, 4, 120, 4, 56, 56, 20, 4, 220, 10, 56, 20, 56, 4, 120, 20, 120, 20, 20, 4, 364, 4, 20, 56, 84, 20, 120, 4, 56

Offset: 1

Felix Huber, Oct 25 2024

nonn

Equivalently, a(n) is the number of unordered triples of divisors of n.

There are tau(n)*(tau(n) - 1)*(tau(n) - 2)/6 distinct cuboids with three different edges, (tau(n)*1*(tau(n) - 1) + tau(n)*(tau(n) - 1)*2)/3 distinct cuboids with two different edges and tau(n) distinct cuboids that are cubes.

			a(4) = 10, because there are 10 distinct cuboids whose edges are divisors of 4: (1, 1, 1), (1, 1, 2), (1, 1, 4), (1, 2, 2), (1, 2, 4), (1, 4, 4), (2, 2, 2), (2, 2, 4), (2, 4, 4), (4, 4, 4).

Felix Huber, Table of n, a(n) for n = 1..10000

Cf. A000005, A086222.

A377304:=proc(n)
   local d;
   d:=NumberTheory:-tau(n);
   return (d^3+3*d^2+2*d)/6
end proc;
seq(A377304(n),n=1..68);

a[n_] := Binomial[DivisorSigma[0, n] + 2, 3]; Array[a, 70] (* Amiram Eldar, Nov 07 2024 *)

a(n) = (tau(n)^3 + 3*tau(n)^2 + 2*tau(n))/6.

a(n) = binomial(tau(n) + 2, 3).

cp's OEIS Frontend

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

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A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.

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A086165 a(n) = |{ (x,y,z) | x < y < z and lcm(x,y,z) = n}|.

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A377304 a(n) is the number of distinct cuboids whose edges are divisors of n.

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Examples

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Programs

Maple

Mathematica

Formula