# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/

Search: id:a368916
Showing 1-1 of 1

%I A368916 #46 Mar 30 2024 17:22:01
%S A368916 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,
%T A368916 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,
%U A368916 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
%N 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.
%C A368916 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.
%C A368916 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
%H A368916 Antti Karttunen, <a href="/A368916/b368916.txt">Table of n, a(n) for n = 0..100000</a>
%H A368916 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A368916 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A368916 a(n) = A368914(A276086(n)) = A368915(A276086(n)).
%F A368916 a(n) = A359550(A327860(n)) = A359550(A342002(n)).
%F A368916 a(n) = 1 - A360111(A276086(n)).
%F A368916 a(n) = [0 == A342019(n)], where [ ] is the Iverson bracket.
%F A368916 For n > 0, a(n) >= A328306(n).
%o A368916 (PARI)
%o A368916 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A368916 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A368916 A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };
%o A368916 A368916(n) = { my(u=A276086(n)); ((u>1)&&A359550(A003415(u))); };
%Y A368916 Characteristic function of A368917 whose complement A342018 gives the positions of zeros after the initial one at a(0).
%Y A368916 Cf. A003415, A048103, A276086, A328306, A341996, A327860, A342002, A342019, A359550, A360111, A368906, A368914, A368915, A368918 (partial sums), A368919, A369653.
%K A368916 nonn
%O A368916 0
%A A368916 _Antti Karttunen_, Jan 09 2024

# Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE