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%I #27 Jun 18 2024 11:08:18
%S 1,0,0,0,1,1,0,0,0,0,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1,0,0,
%T 1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1,
%U 0,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1,0,0,1,1,1,0,0,0,1,1,0,0,0,0,1,1,0,1
%N a(n) = 1 if A327860(n) is a multiple of 3, otherwise 0, where A327860 is the arithmetic derivative of the primorial base exp-function.
%C Conjecture: asymptotic mean is 4/9: One third of the primorial base representations (A049345 from which the intermediate value x = A276086(n) is built from) have "2" as their second rightmost digit (out of 0, 1, 2). That 2 is converted by A276086 to 3^2, so as x is a multiple of nine, it is guaranteed that x' [= A003415(x)] gives a multiple of three. If the second rightmost digit is "1", then x = A276086(n) is a multiple of 3 (but not of 9), and its arithmetic derivative, x' = 3*(x/3)' + (x/3), certainly is not a multiple of 3. This leaves the case where the second rightmost digit is "0", and conjecturing now in this case that the modulo 3 remainders are eventually evenly distributed for x' (see A359430 and A373256), at least among the number of the form A048103, it gives in this case 1/3 * 1/3 = 1/9 probability that x' is a multiple of 3, so the total probability would be 1/3 + 0 + 1/9 = 4/9. Note that Sum_{i=1..10^n} a(i), for n = 1..8 gives: 3, 43, 443, 4443, 44443, 444443, 4444444, 44444444, so empirically the value 4/9 seems very sharp. - revised Jun 18 2024.
%C Compare also to A369034 and A373601.
%H Antti Karttunen, <a href="/A369653/b369653.txt">Table of n, a(n) for n = 0..65537</a>
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F a(n) = A079978(A327860(n)) = A359430(A276086(n)).
%F If n == 0 or 1 (mod 6), a(n) = A373601(n), if n == 2 or 3 (mod 6), a(n) = 0, and if n == 4 or 5 (mod 6), a(n) = 1. - _Antti Karttunen_, Jun 18 2024
%o (PARI)
%o A327860(n) = { my(s=0, m=1, p=2, e); while(n, e = (n%p); m *= (p^e); s += (e/p); n = n\p; p = nextprime(1+p)); (s*m); };
%o A369653(n) = !(A327860(n)%3);
%o (PARI) A369653(n) = { my(p=2, c1=0, c2=0); while(n, if(3==p, if((n%p), return(2==(n%p))), if(1==(p%3), c1 += (n%p), if(2==(p%3), c2 += (n%p)))); n = n\p; p = nextprime(1+p)); 0==((c1-c2)%3); }; \\ _Antti Karttunen_, Jun 18 2024
%Y Characteristic function of A369654.
%Y Cf. A048103, A079978, A276086, A327860.
%Y Cf. also A359430, A369034, A369643, A373256, A373598, A373601.
%K nonn
%O 0
%A _Antti Karttunen_, Jan 28 2024
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