A369001 - OEIS

A369001

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

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

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

Antti Karttunen, Table of n, a(n) for n = 1..65537

Index entries for characteristic functions

FORMULA

a(n) = A059841(A083345(n)).

a(n) >= A369004(n).

A353557(n) <= a(n) <= A368997(n) <= A358680(n).

From Antti Karttunen, Jan 29 2024 and Feb 08 2024: (Start)

For n > 1, a(n) = [A007814(A003415(n)) > A007814(n)], where [ ] is the Iverson bracket.

a(n) = A353557(n) + [A007814(n)==2]*A066829(n) + [A007814(n)>=4]*a(n/16).

a(n) = a(16*n).

a(n) <= A035263(n).

a(n) = A121262(A276085(n)). [See comments]

(End)

a(n) = A035263(n) - A373141(n) = A369004(n) + A373264(n) = A373137(n) + A373266(n). - Antti Karttunen, Jun 01 2024

PROG

(PARI)

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

A369001(n) = !(A083345(n)%2);

(PARI)

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

A369001(n) = ((1==n)||(valuation(A003415(n), 2) > valuation(n, 2)));

(PARI) A369001(n) = (((n%2)&&(!(bigomega(n)%2))) || (2==valuation(n, 2) && (bigomega(n)%2)) || (valuation(n, 2)>=4 && A369001(n/16)));

CROSSREFS

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, A373137, A373141, A373264, A373266.

Cf. A369974 (Dirichlet inverse), A369975 (and its parity), A369980 (run lengths).

Cf. also A358680, A368997, A369643, A372573.

Sequence in context: A324917 A369974 A369975 * A361024 A354037 A185118

Adjacent sequences: A368998 A368999 A369000 * A369002 A369003 A369004

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 14 2024

STATUS

approved