A369974 - OEIS

%I #9 Feb 10 2024 00:08:12

%S 1,0,0,0,0,0,0,0,-1,0,0,-1,0,0,-1,-1,0,0,0,-1,-1,0,0,0,-1,0,0,-1,0,0,

%T 0,0,-1,0,-1,0,0,0,-1,0,0,0,0,-1,0,0,0,0,-1,0,-1,-1,0,0,-1,0,-1,0,0,0,

%U 0,0,0,0,-1,0,0,-1,-1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,-1,0,-1,0,0,0,-1,-1,-1,0,-1,0

%N Dirichlet inverse of A369001, where A369001(n) = 1 if n' / gcd(n,n') is even, otherwise 0, and n' stands for the arithmetic derivative of n, A003415.

%C a(144) = 2 is the first term > 1.

%H Antti Karttunen, <a href="/A369974/b369974.txt">Table of n, a(n) for n = 1..65537</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A369001(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 A369001(n) = !(A083345(n)%2);

%o memoA369974 = Map();

%o A369974(n) = if(1==n,1,my(v); if(mapisdefined(memoA369974,n,&v), v, v = -sumdiv(n,d,if(d<n,A369001(n/d)*A369974(d),0)); mapput(memoA369974,n,v); (v)));

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

%Y Agrees paritywise with A369978.

%Y Cf. A358777, A359763, A359773, A359780 for similar sequences.

%K sign

%O 1,144

%A _Antti Karttunen_, Feb 09 2024