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a(n) = 1 if A048675(n) is a multiple of 3, otherwise 0.
16

%I #20 Apr 16 2022 09:22:01

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

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

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

%N a(n) = 1 if A048675(n) is a multiple of 3, otherwise 0.

%C Proof for the formula a(n) = A353269(A332462(n)): This follows from a more general identity A048675(n) == A156552(A019565(A156552(n))) (mod 3). Applying A156552's inverse, the Doudna-map x -> A005940(1+x), to the both sides from the right we obtain A087808(n) == A048678(n) (mod 3), and from the respective recurrences of those two sequences it is easy to see that they are equal when reduced modulo 3.

%H Antti Karttunen, <a href="/A353350/b353350.txt">Table of n, a(n) for n = 1..65537</a>

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

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A079978(A048675(n)).

%F a(n) = A353269(A332462(n)). [See comments for a proof]

%F a(n) = 1 iff A332813(n) = 0, or equally iff A332823(n) = 0.

%F a(p) = 0 for all primes p.

%F a(n) = a(A003961(n)) = a(A348717(n)), for all n >= 1.

%t f[p_, e_] := e*2^(PrimePi[p] - 1); a[1] = 1; a[n_] := Boole @ Divisible[Plus @@ f @@@ FactorInteger[n], 3]; Array[a, 100] (* _Amiram Eldar_, Apr 15 2022 *)

%o (PARI)

%o A048675(n) = { my(f = factor(n)); sum(k=1, #f~, f[k, 2]*2^primepi(f[k, 1]))/2; };

%o A353350(n) = (0==(A048675(n)%3));

%Y Characteristic function of A332820.

%Y Cf. A003961, A048675, A079978, A332462, A332813, A332823, A348717,

%Y Cf. A353348 (Dirichlet inverse), A353349 (sum with it), A353352 (inverse Möbius transform), A353353.

%Y Cf. also A353269, A353380.

%K nonn

%O 1

%A _Antti Karttunen_, Apr 15 2022