cp's OEIS Frontend

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

%I A070919 #46 Sep 03 2023 08:45:05
%S A070919 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,
%T A070919 49,37,133,7,343,7,91,49,49,49,361,7,49,49,259,7,343,7,133,133,49,7,
%U A070919 427,19,133,49,133,7,259,49,259,49,49,7,931,7,49,133,127,49,343,7,133
%N A070919 a(n) = Card{ (x,y,z) | lcm(x,y,z)=n }.
%C A070919 A048691(n) gives Card{ (x,y) | lcm(x,y)=n }.
%H A070919 Antti Karttunen, <a href="/A070919/b070919.txt">Table of n, a(n) for n = 1..10000</a>
%H A070919 O. Bagdasar, <a href="https://doi.org/10.5937/SPSUNP1402091B">On some functions involving the lcm and gcd of integer tuples</a>, Scientific Publications of the State University of Novi Pazar, Appl. Maths. Inform. and Mech., Vol. 6, 2 (2014), 91-100.
%F A070919 a(n) = Sum_{d|n} A000005(d)^3*A008683(n/d).
%F A070919 Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^3/k^s.
%F A070919 Multiplicative with a(p^e) = 1+3*e+3*e^2 for prime p and e >= 0. - _Werner Schulte_, Nov 30 2018
%t A070919 Join[{1},Table[Product[(k + 1)^3 - k^3, {k, FactorInteger[n][[All, 2]]}], {n,2, 68}]] (* _Geoffrey Critzer_, Jan 10 2015 *)
%o A070919 (PARI) for(n=1,100,print1(sumdiv(n,d,numdiv(d)^3*moebius(n/d)),","))
%o A070919 (PARI) a(n) = vecprod(apply(x->(x+1)^3-x^3, factor(n)[, 2])); \\ _Amiram Eldar_, Sep 03 2023
%Y A070919 Cf. A000005, A008683, A048691, A070920, A070921, A086222, A247513.
%K A070919 mult,easy,nonn
%O A070919 1,2
%A A070919 _Benoit Cloitre_, May 20 2002