A369658 a(n) = 1 if n is not multiple of 3, but its arithmetic derivative is, otherwise 0.

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

Offset

0

Comments

Conjecture: the asymptotic mean of this sequence is (2/3)*(1/3) = 2/9. Compare to the comment at A369653, but consider also the four lowermost rows of the table given at A369252 (and further generalizations to various number of primes), and also A007352, A096629, and how they affect such probabilities.

Sum_{i=1..10^n} a(i), for n = 1..10 gives: 2, 18, 201, 2110, 21484, 216973, 2181521, 21896827, 219541804, 2199637607. - Antti Karttunen, Jun 17 2024

Links

Antti Karttunen, Table of n, a(n) for n = 0..100000

Eric Weisstein's World of Mathematics, Chebyshev Bias

Index entries for characteristic functions

Index entries for sequences which agree for a long time but are different

Formula

a(n) = A011655(n) * A079978(A003415(n)) = A011655(n) * A359430(n).

a(n) <= A369643(n) <= A359430(n).

For n >= 1, a(n) <= A373474(n). - Antti Karttunen, Jun 07 2024

For n >= 1, a(n) = A011655(n) * A373371(n) = A011655(n) * [A373591(n) == A373592(n) (mod 3)]. - Antti Karttunen, Jun 13 2024

Prog

(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A369658(n) = ((n%3)&&(0==(A003415(n)%3)));
(PARI) A369658(n) = if(n<2, n, if(!(n%3), 0, my(f = factor(n), m1=0, m2=0); for(i=1, #f~, if(1==(f[i, 1]%3), m1 += f[i, 2], if(2==(f[i, 1]%3), m2 += f[i, 2]))); 0==((m1-m2)%3))); \\ Antti Karttunen, Jun 16 2024

Crossrefs

Characteristic function of A369659.

Cf. A003415, A011655, A079978, A359430, A369653, A373371, A373591, A373592.

Cf. A007352, A096629, A369252.

Differs from related A369643 for the first time at n=54, where a(54) = 0, while A369643(54) = 1.

Differs from related A373474 for the first time at n=19683, where a(19683) = 0, while A373474(19683) = 1.

Cf. also A353557, A360109, A369968, for cases k = 2, 4, 5 of the characteristic functions for nonmultiples of k whose arithmetic derivative is multiple of k.

Sequence in context: A098108 A363712 A030214 * A025464 A373260 A162518

Adjacent sequences: A369655 A369656 A369657 * A369659 A369660 A369661

Keyword

nonn

Author

Antti Karttunen, Feb 10 2024

Status

approved