A369004 - OEIS

%I #14 Jan 15 2024 10:02:51

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

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

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

%N a(n) = 1 if n' / gcd(n,n') is a multiple of 4, otherwise 0, where n' stands for the arithmetic derivative of n, A003415(n).

%C a(n) = 1 if A083345(n) = the numerator of Sum(e/p: n=Product(p^e)) is of the form 4k, and 0 if it is not.

%C Question: Is the asymptotic mean of this sequence 1/6? See also A369001.

%H Antti Karttunen, <a href="/A369004/b369004.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) = A121262(A083345(n)).

%F a(n) <= A369001(n).

%F a(n) <= A368994(n) <= A353494(n).

%F a(n) <= 1 - A369006(n).

%o (PARI)

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

%o A369004(n) = !(A083345(n)%4);

%Y Characteristic function of A369005.

%Y Cf. A083345, A121262, A369001, A369006.

%Y Cf. also A353494, A368994.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 14 2024