%I #9 Feb 09 2024 15:49:39

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

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

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

%N Parity of A369974.

%H Antti Karttunen, <a href="/A369975/b369975.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(n) = A369974(n) mod 2 = A369978(n) mod 2.

%F a(n) <= A369001(n). [Because A369002 is a multiplicative semigroup; proof is similar to one given in A359780.]

%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)));

%o A369975(n) = (A369974(n)%2);

%Y Characteristic function of A369976.

%Y Parity of A369974 and of A369978.

%Y Cf. A083345, A369002.

%Y Differs from A369001 for the first time at n=81, where a(81) = 0, while A369001(81) = 1.

%Y Cf. also A359780.

%K nonn

%O 1

%A _Antti Karttunen_, Feb 09 2024