A369658 - OEIS

%I #30 Jun 18 2024 08:19:05

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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

%H Antti Karttunen, <a href="/A369658/b369658.txt">Table of n, a(n) for n = 0..100000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChebyshevBias.html">Chebyshev Bias</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%H <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>

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

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

%F For n >= 1, a(n) <= A373474(n). - _Antti Karttunen_, Jun 07 2024

%F For n >= 1, a(n) = A011655(n) * A373371(n) = A011655(n) * [A373591(n) == A373592(n) (mod 3)]. - _Antti Karttunen_, Jun 13 2024

%o (PARI)

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

%o A369658(n) = ((n%3)&&(0==(A003415(n)%3)));

%o (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

%Y Characteristic function of A369659.

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

%Y Cf. A007352, A096629, A369252.

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

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

%Y 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.

%K nonn

%O 0

%A _Antti Karttunen_, Feb 10 2024