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%I #31 Apr 18 2024 12:50:18
%S 1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T 0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%U 0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,1
%N a(n) = 1 if A327860(n) is a multiple of 4, otherwise 0, where A327860 is the arithmetic derivative of the primorial base exp-function.
%C Asymptotic mean is most likely 1/8, but with a caveat similar to A358846 (consider how Chebyshev's bias, A038698, affects this and see comments in A235992, A327860, and especially in A360110. Note that A276086 never produces multiples of 4, so only way A327860(n) can be an even number is when A276086(n) is an odd number with an even number of prime factors with multiplicity (A046337), which set has the density 1/4. Assuming that for x = A276086(4*n), the arithmetic derivative x' is then evenly distributed between numbers of the form 4k and 4k+2, we get 1/4 * 1/2 = 1/8.
%C Compare to the empirical asymptotic means of A368994 and A369004, and also to A369653.
%H Antti Karttunen, <a href="/A369034/b369034.txt">Table of n, a(n) for n = 0..100000</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) = A121262(A327860(n)) = [A353630(n) == 0], where [ ] is the Iverson bracket.
%F a(n) = A353494(A276086(n)) = A368994(A276086(n)) = A369004(A276086(n)).
%F a(n) = A121262(n) - A369036(n).
%F a(n) = A360109(A276086(n)). - _Antti Karttunen_, Apr 13 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 A369034(n) = !(A327860(n)%4);
%Y Characteristic function of A369035.
%Y Cf. A038698, A121262, A276086, A327860, A353494, A353630, A358846, A360109, A360110, A368994, A369004, A369036, A369653.
%Y Differs from A342019 for the first time at n=126, where a(126) = 0, while A342019(126) = 1.
%K nonn
%O 0
%A _Antti Karttunen_, Jan 20 2024
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