A373155 - OEIS

%I #13 May 29 2024 07:05:29

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

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

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

%N a(n) = 1 if n is a non-multiple of 3 whose 2-adic valuation is even, otherwise 0.

%H Antti Karttunen, <a href="/A373155/b373155.txt">Table of n, a(n) for n = 1..100000</a>

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

%F Multiplicative with a(2^e) = 1 if e is even, 0 if e is odd, a(3^e) = 0, and a(p^e) = 1 for any prime p > 3.

%F a(n) = A011655(n) * A035263(n).

%F a(n) = abs(A084091(n)) = A084091(n) mod 2.

%F From _Amiram Eldar_, May 29 2024: (Start)

%F Dirichlet g.f.: (2^s/(2^s+1)) * (1-1/3^s) * zeta(s).

%F Sum_{k=1..n} a(k) ~ (4/9) * n. (End)

%t a[n_] := If[EvenQ[IntegerExponent[n, 2]] && !Divisible[n, 3], 1, 0]; Array[a, 100] (* _Amiram Eldar_, May 29 2024 *)

%o (PARI) A373155(n) = ((n%3) && !(valuation(n,2)%2));

%o (PARI) A373155(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k, 1], !(f[k, 2]%2), f[k, 1] > 3)); };

%Y Characteristic function of A084087.

%Y Absolute values and also parity of A084091.

%Y Cf. A007814, A011655, A035263.

%K nonn,mult,easy

%O 1

%A _Antti Karttunen_, May 28 2024