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Dirichlet inverse of sequence b(n) = 1+A083345(n), where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.
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%I #11 Feb 10 2024 00:08:07

%S 1,-2,-2,2,-2,2,-2,-4,1,0,-2,3,-2,-2,-1,9,-2,4,-2,9,-3,-6,-2,-8,1,-8,

%T 2,15,-2,12,-2,-18,-7,-12,-5,-14,-2,-14,-9,-22,-2,18,-2,27,10,-18,-2,

%U 20,1,10,-13,33,-2,-8,-9,-36,-15,-24,-2,-16,-2,-26,14,36,-11,30,-2,45,-19,16,-2,22,-2,-32,12,51,-11,36

%N Dirichlet inverse of sequence b(n) = 1+A083345(n), where A083345(n) = n' / gcd(n,n'), and n' stands for the arithmetic derivative of n, A003415.

%H Antti Karttunen, <a href="/A369978/b369978.txt">Table of n, a(n) for n = 1..16384</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} (1+A083345(n/d)) * a(d).

%o (PARI)

%o A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

%o memoA369978 = Map();

%o A369978(n) = if(1==n,1,my(v); if(mapisdefined(memoA369978,n,&v), v, v = -sumdiv(n,d,if(d<n,(1+A083345(n/d))*A369978(d),0)); mapput(memoA369978,n,v); (v)));

%Y Cf. A003415, A083345, A369001, A369974, A369975 (parity of terms), A369976 (positions of odd terms).

%Y Cf. A359790 and A366265 for similar sequences.

%K sign

%O 1,2

%A _Antti Karttunen_, Feb 09 2024