A086222 - OEIS

%I #18 Sep 03 2023 08:44:52

%S 1,3,3,6,3,13,3,10,6,13,3,30,3,13,13,15,3,30,3,30,13,13,3,54,6,13,10,

%T 30,3,71,3,21,13,13,13,73,3,13,13,54,3,71,3,30,30,13,3,85,6,30,13,30,

%U 3,54,13,54,13,13,3,178,3,13,30,28,13,71,3,30,13,71,3,135,3,13,30,30,13,71,3

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

%C A number of conjectures are possible, many of which should be easy to prove. Examples: (1) If n is a product of two primes then a(n)=13. (2) If n is a square of a prime then a(n)=6. - _John W. Layman_, Sep 01 2003

%H Antti Karttunen, <a href="/A086222/b086222.txt">Table of n, a(n) for n = 1..10000</a>

%F For a prime p, a(p) = 3.

%F a(n) = (A070919(n) + 3*A048691(n) + 2)/6. - _Vladeta Jovovic_, Dec 01 2004

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

%o (PARI)

%o A048691(n) = numdiv(n^2);

%o A070919(n) = sumdiv(n, d, (numdiv(d)^3)*moebius(n/d));

%o A086222(n) = ((A070919(n)+3*A048691(n)+2)/6); \\ _Antti Karttunen_, May 19 2017, after Jovovic's formula.

%o (PARI) 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

%Y Cf. A048691, A070919, A018892, A086165.

%K nonn,easy

%O 1,2

%A Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003

%E More terms from _John W. Layman_, Sep 01 2003