OFFSET
1
COMMENTS
a(n) = 1 if A083345(n) = numerator of Sum(e/p: n=Product(p^e)) is even, and 0 if it is odd.
Question: Is the asymptotic mean of this sequence 1/3? See also A369004.
Answer to the above question is yes, as 1/4 + 1/16 + 1/64 + 1/256 + 1/1024 + 1/4096 + ... = 1/3. See the new recursive formula, whose first term contributes 1/4, and the second term 1/16 to the total asymptotic mean, with the rest obtained by recursion. For a proof, consider A001787(n) = A003415(2^n) = n*2^(n-1). We have A007814(A001787(n)) > n iff n is a multiple of 4. - Antti Karttunen, Jan 29 2024
Also a(n) = 1 iff A276085(n) is a multiple of 4. See comment in A327860, which applies also to A342002 [= A083345(A276086(n))]. Therefore, A121262(n) = A059841(A342002(n)) = A059841(A083345(A276086(n))) = a(A276086(n)). Assuming that the new formula a(n) = A121262(A276085(n)) holds, then substituing A276086(n) for n in it gives us back equation a(A276086(n)) = A121262(n) that was proved above. - Antti Karttunen, Feb 08 2024
LINKS
FORMULA
a(n) >= A369004(n).
From Antti Karttunen, Jan 29 2024 and Feb 08 2024: (Start)
a(n) = a(16*n).
a(n) <= A035263(n).
(End)
PROG
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Jan 14 2024
STATUS
approved