A369034 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.

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, 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, 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

Offset

0

Comments

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.

Compare to the empirical asymptotic means of A368994 and A369004, and also to A369653.

Links

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

Index entries for characteristic functions

Index entries for sequences related to primorial base

Formula

a(n) = A121262(A327860(n)) = [A353630(n) == 0], where [ ] is the Iverson bracket.

a(n) = A353494(A276086(n)) = A368994(A276086(n)) = A369004(A276086(n)).

a(n) = A121262(n) - A369036(n).

a(n) = A360109(A276086(n)). - Antti Karttunen, Apr 13 2024

Prog

(PARI)
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); };
A369034(n) = !(A327860(n)%4);

Crossrefs

Characteristic function of A369035.

Cf. A038698, A121262, A276086, A327860, A353494, A353630, A358846, A360109, A360110, A368994, A369004, A369036, A369653.

Differs from A342019 for the first time at n=126, where a(126) = 0, while A342019(126) = 1.

Sequence in context: A104124 A347246 A052434 * A015241 A253513 A014025

Adjacent sequences: A369031 A369032 A369033 * A369035 A369036 A369037

Keyword

nonn

Author

Antti Karttunen, Jan 20 2024

Status

approved