A353335 - OEIS

%I #9 Apr 20 2022 15:47:07

%S 1,-1,-2,0,-3,2,-4,0,-5,3,-5,0,-6,4,0,0,-7,5,-8,0,-3,5,-10,0,-8,6,-14,

%T 0,-11,0,-13,0,-2,7,-2,0,-14,8,-5,0,-15,3,-16,0,7,10,-18,0,-25,8,-4,0,

%U -20,14,-1,0,-7,11,-21,0,-23,13,8,0,-4,2,-24,0,-9,2,-25,0,-27,14,4,0,-8,5,-28,0,-52,15,-30,0,-3

%N Dirichlet inverse of A353420.

%H Antti Karttunen, <a href="/A353335/b353335.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(1) = 1; a(n) = -Sum_{d|n, d < n} A353420(n/d) * a(d).

%F a(n) = A353336(n) - A353420(n).

%o (PARI)

%o up_to = 65537;

%o DirInverseCorrect(v) = { my(u=vector(#v)); u[1] = (1/v[1]); for(n=2, #v, u[n] = (-u[1]*sumdiv(n, d, if(d<n, v[n/d]*u[d], 0)))); (u) }; \\ Compute the Dirichlet inverse of the sequence given in input vector v (correctly!)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A126760(n) = {n&&n\=3^valuation(n, 3)<<valuation(n, 2); n%3+n\6*2}; \\ From A126760

%o A353420(n) = A126760(A003961(n));

%o v353335 = DirInverseCorrect(vector(up_to,n,A353420(n)));

%o A353335(n) = v353335[n];

%Y Cf. A003961, A126760, A353420, A353336.

%K sign

%O 1,3

%A _Antti Karttunen_, Apr 20 2022