A354874 -id:A354874

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A368915

a(n) = 1 if there is no prime p such that p^p divides the arithmetic derivative of n, and 0 otherwise.

+10
9

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1

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

OFFSET

COMMENTS

Question: What is the asymptotic mean of this sequence (and its complement A341996)? Knowing the value for A360111 would solve this. See also related sequences like A354874 and A368916.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(1) = 0; for n > 1, a(n) = A359550(A003415(n)).

For all n > 1, a(n) = 1 - A341996(n) = A359550(n) - A360111(n).

For all n > 1, A359550(n) >= a(n) >= A328308(n).

For all n >= 1, a(n) >= A354874(n).

a(n) = A368914(n) - A368913(n).

PROG

(PARI)

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

A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

A368915(n) = ((n>1)&&A359550(A003415(n)));

CROSSREFS

Characteristic function of A358215.

Cf. A003415, A328308, A341996 (one's complement), A354874, A359550, A360111, A368913, A368914, A368916 [= a(A276086(n))].

KEYWORD

nonn

AUTHOR

Antti Karttunen, Jan 09 2024

STATUS

approved

A369669

The greatest common divisor of the first and the second arithmetic derivative of n.

+10
4

0, 0, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 16, 1, 3, 4, 16, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 27, 16, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 16, 4, 1, 1, 1, 16, 1, 5, 1, 16, 1, 3, 4, 4, 1, 27, 16, 4, 1, 1, 1, 4, 1, 1, 1, 64, 3, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 16, 3, 1, 1, 16, 108, 1, 1, 4, 1, 3, 16, 4, 1, 1, 4, 16, 1, 7, 4, 16, 1, 1, 5

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

OFFSET

0,5

LINKS

Antti Karttunen, Table of n, a(n) for n = 0..16384

FORMULA

a(n) = gcd(A003415(n), A068346(n)).

For n >= 2, a(n) = A085731(A003415(n)).

PROG

(PARI)

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

A369669(n) = { my(d=A003415(n)); gcd(d, A003415(d)); };

CROSSREFS

Cf. A003415, A068346, A085731, A370130 [= a(A276086(n))].

Cf. A328393 (positions of 1's), A354874 (their characteristic function).

Cf. A327864 (positions of even terms, also positions of multiples of 4).

Cf. A370119 (positions of multiples of 3).

KEYWORD

nonn

AUTHOR

Antti Karttunen, Feb 10 2024

STATUS

approved

A368912

a(n) = 1 if A342001(n) is squarefree, and 0 otherwise.

+10
3

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1

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

OFFSET

COMMENTS

Question: What is the asymptotic mean of this and related sequences like A354874 and A368914?

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..100000

Index entries for characteristic functions

FORMULA

a(1) = 0, and for n > 1, a(n) = A008966(A342001(n)).

For all n >= 1, A354874(n) <= a(n) <= A368914(n).

PROG

(PARI)