cp's OEIS Frontend

A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).

%I A347395 #13 Mar 04 2023 05:21:25
%S A347395 0,1,1,1,1,3,1,2,1,5,1,3,1,7,6,2,1,2,1,5,8,11,1,5,1,13,2,7,1,14,1,3,
%T A347395 12,17,10,2,1,19,14,9,1,20,1,11,5,23,1,5,1,2,18,13,1,4,14,13,20,29,1,
%U A347395 14,1,31,7,3,16,32,1,17,24,34,1,3,1,37,3,19,16,38,1,9,2,41,1,20,20,43,30,21,1,9,18,23,32
%N A347395 Dirichlet convolution of Liouville's lambda (A008836) with A342001, where A342001(n) = A003415(n)/A003557(n).
%C A347395 It seems that all the terms after the initial zero are strictly positive. Checked up to n = 2^24. Compare to A346485.
%H A347395 Antti Karttunen, <a href="/A347395/b347395.txt">Table of n, a(n) for n = 1..16384</a>
%H A347395 Antti Karttunen, <a href="/A347395/a347395.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A347395 a(n) = Sum_{d|n} A008836(n/d) * A342001(d).
%F A347395 Sum_{k=1..n} a(k) ~ c * A065464 * Pi^4 * n^2 / 180, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - _Vaclav Kotesovec_, Mar 04 2023
%o A347395 (PARI)
%o A347395 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A347395 A003557(n) = (n/factorback(factorint(n)[, 1]));
%o A347395 A342001(n) = (A003415(n) / A003557(n));
%o A347395 A008836(n) = ((-1)^bigomega(n));
%o A347395 A347395(n) = sumdiv(n,d,A008836(n/d)*A342001(d));
%Y A347395 Cf. A003415, A003557, A008836, A342001, A347396 [= a(A276086(n))].
%Y A347395 Cf. also A346485, A347235.
%K A347395 nonn,look
%O A347395 1,6
%A A347395 _Antti Karttunen_, Sep 02 2021