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A369658
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a(n) = 1 if n is not multiple of 3, but its arithmetic derivative is, otherwise 0.
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11
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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
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OFFSET
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0
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COMMENTS
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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
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LINKS
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FORMULA
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PROG
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(PARI)
A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
(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
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CROSSREFS
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Characteristic function of A369659.
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.
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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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