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A359790 Dirichlet inverse of function f(n) = 1 + n', where n' stands for the arithmetic derivative of n, A003415(n).

%I A359790 #12 Jan 17 2023 10:01:24
%S A359790 1,-2,-2,-1,-2,2,-2,-1,-3,0,-2,3,-2,-2,-1,0,-2,6,-2,3,-3,-6,-2,7,-7,
%T A359790 -8,-8,3,-2,12,-2,3,-7,-12,-5,9,-2,-14,-9,11,-2,18,-2,3,0,-18,-2,11,
%U A359790 -11,6,-13,3,-2,26,-9,15,-15,-24,-2,17,-2,-26,-4,9,-11,30,-2,3,-19,16,-2,9,-2,-32,0,3,-11,36,-2,23,-16,-36
%N A359790 Dirichlet inverse of function f(n) = 1 + n', where n' stands for the arithmetic derivative of n, A003415(n).
%H A359790 Antti Karttunen, <a href="/A359790/b359790.txt">Table of n, a(n) for n = 1..65537</a>
%F A359790 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A003415(n/d)) * a(d).
%o A359790 (PARI)
%o A359790 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A359790 memoA359790 = Map();
%o A359790 A359790(n) = if(1==n,1,my(v); if(mapisdefined(memoA359790,n,&v), v, v = -sumdiv(n,d,if(d<n,(1+A003415(n/d))*A359790(d),0)); mapput(memoA359790,n,v); (v)));
%Y A359790 Cf. A003415, A359780, A359781 (parity of terms), A359782 (positions of even terms), A359783 (of odd terms).
%Y A359790 Cf. also A346241, A347082, A347084, A359603, A359789, A359791 [= a(A003961(n))] (for similar constructions).
%K A359790 sign
%O A359790 1,2
%A A359790 _Antti Karttunen_, Jan 13 2023