A368916 - OEIS

A368916

a(n) = 1 if there is no prime p such that p^p divides the arithmetic derivative of A276086(n), and 0 otherwise, where A276086 is the primorial base exp-function.

0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1

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OFFSET

COMMENTS

Question: What is the asymptotic mean of this sequence (equal to the density of A368914 or A368915 when restricted to the terms of A048103)? Straightness of the graphs of A368917, A368918 and A368906 seems to indicate that the expected value of A368914(n) is not correlated with the specific features (like the magnitude of its exponents) in the prime factorization of n.

Answer: If the conjecture that "For any given prime p, the distribution of the p-adic valuations of n' among those n that are not multiples of p, is the same as the distribution of the p-adic valuations among all the natural numbers", then the asymptotic mean of this sequence is equal to Product_{p prime} (1 - 1/p^(1+p)) = 0.864142072322... See also A368919, A360111 and A369653. - Antti Karttunen, Jan 29 and Feb 10 2024

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) = A368914(A276086(n)) = A368915(A276086(n)).

a(n) = A359550(A327860(n)) = A359550(A342002(n)).

a(n) = 1 - A360111(A276086(n)).

a(n) = [0 == A342019(n)], where [ ] is the Iverson bracket.

For n > 0, a(n) >= A328306(n).

PROG

(PARI)