A346485 - OEIS

%I #22 Mar 04 2023 05:08:34

%S 0,1,1,1,1,3,1,1,1,5,1,2,1,7,6,1,1,1,1,4,8,11,1,2,1,13,1,6,1,14,1,1,

%T 12,17,10,0,1,19,14,4,1,20,1,10,4,23,1,2,1,1,18,12,1,1,14,6,20,29,1,8,

%U 1,31,6,1,16,32,1,16,24,34,1,0,1,37,2,18,16,38,1,4,1,41,1,12,20,43,30,10,1,4,18,22,32,47

%N Möbius transform of A342001, where A342001(n) = A003415(n)/A003557(n).

%C Conjecture 1: After the initial zero, the positions of other zeros is given by A036785.

%C Conjecture 2: No negative terms. Checked up to n = 2^24.

%H Antti Karttunen, <a href="/A346485/b346485.txt">Table of n, a(n) for n = 1..10000</a>

%H Antti Karttunen, <a href="/A346485/a346485.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = Sum_{d|n} A008683(n/d) * A342001(d).

%F Dirichlet g.f.: Product_{p prime} (1+p^(1-s)-p^(-s)) * Sum_{p prime} p^s/((p^s-1)*(p^s+p-1)). - _Sebastian Karlsson_, May 08 2022

%F Sum_{k=1..n} a(k) ~ c * A065464 * n^2 / 2, where c = Sum_{j>=2} (1/2 + (-1)^j * (Fibonacci(j) - 1/2))*PrimeZetaP(j) = 0.4526952873143153104685540856936425315834753528741817723313791528384... - _Vaclav Kotesovec_, Mar 04 2023

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A003557(n) = (n/factorback(factorint(n)[, 1]));

%o A342001(n) = (A003415(n) / A003557(n));

%o A346485(n) = sumdiv(n,d,moebius(n/d)*A342001(d));

%Y Cf. A003415, A003557, A008683, A036785, A342001, A347232 [= a(A276086(n))].

%Y Cf. also A300251, A300717, A347234, A347235, A347395.

%K nonn

%O 1,6

%A _Antti Karttunen_, Aug 26 2021