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A369001
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a(n) = 1 if n' / gcd(n,n') is even, otherwise 0, where n' stands for the arithmetic derivative of n, A003415.
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12
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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, 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, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 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
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OFFSET
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1
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COMMENTS
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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
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LINKS
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FORMULA
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a(n) = a(16*n).
(End)
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PROG
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(PARI)
A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };
(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
(PARI) A369001(n) = (((n%2)&&(!(bigomega(n)%2))) || (2==valuation(n, 2) && (bigomega(n)%2)) || (valuation(n, 2)>=4 && A369001(n/16)));
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CROSSREFS
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Characteristic function of A369002, whose complement A369003 gives the positions of 0's.
Cf. A001787, A003415, A007814, A035263, A059841, A066829, A083345, A085731, A121262 [= a(A276086(n))], A276085, A276086, A327860, A342002, A353557, A359820, A369004, A369643.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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