A070919 - OEIS

%I #46 Sep 03 2023 08:45:05

%S 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 49,37,133,7,343,7,91,49,49,49,361,7,49,49,259,7,343,7,133,133,49,7,

%U 427,19,133,49,133,7,259,49,259,49,49,7,931,7,49,133,127,49,343,7,133

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

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

%H Antti Karttunen, <a href="/A070919/b070919.txt">Table of n, a(n) for n = 1..10000</a>

%H 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 a(n) = Sum_{d|n} A000005(d)^3*A008683(n/d).

%F Sum_{k>0} a(k)/k^s = (1/zeta(s))*Sum_{k>0} tau(k)^3/k^s.

%F Multiplicative with a(p^e) = 1+3*e+3*e^2 for prime p and e >= 0. - _Werner Schulte_, Nov 30 2018

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

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

%o (PARI) a(n) = vecprod(apply(x->(x+1)^3-x^3, factor(n)[, 2])); \\ _Amiram Eldar_, Sep 03 2023

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

%K mult,easy,nonn

%O 1,2

%A _Benoit Cloitre_, May 20 2002