A368915 - OEIS

%I #33 Jan 14 2024 17:32:17

%S 0,1,1,0,1,1,1,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1,1,1,0,1,1,

%T 0,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,1,1,0,1,1,1,0,1,1,1,0,

%U 1,1,1,0,1,1,1,0,1,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,1,1,1,0,1,1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1

%N a(n) = 1 if there is no prime p such that p^p divides the arithmetic derivative of n, and 0 otherwise.

%C Question: What is the asymptotic mean of this sequence (and its complement A341996)? Knowing the value for A360111 would solve this. See also related sequences like A354874 and A368916.

%H Antti Karttunen, <a href="/A368915/b368915.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>

%F a(1) = 0; for n > 1, a(n) = A359550(A003415(n)).

%F For all n > 1, a(n) = 1 - A341996(n) = A359550(n) - A360111(n).

%F For all n > 1, A359550(n) >= a(n) >= A328308(n).

%F For all n >= 1, a(n) >= A354874(n).

%F a(n) = A368914(n) - A368913(n).

%o (PARI)

%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));

%o A359550(n) = { my(f = factor(n)); prod(k=1, #f~, (f[k, 2]<f[k, 1])); };

%o A368915(n) = ((n>1)&&A359550(A003415(n)));

%Y Characteristic function of A358215.

%Y Cf. A003415, A328308, A341996 (one's complement), A354874, A359550, A360111, A368913, A368914, A368916 [= a(A276086(n))].

%K nonn

%O 1

%A _Antti Karttunen_, Jan 09 2024