A369006 - OEIS

%I #17 Jan 15 2024 10:04:35

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

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

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

%N a(n) = 1 if there is no prime p such that p^p divides n' / gcd(n,n'), and 0 otherwise, where n' stands for the arithmetic derivative of n, A003415(n).

%C Question: Is there a formula for the asymptotic mean, which seems to be around 0.813...? Consider A369004 and A369007.

%H Antti Karttunen, <a href="/A369006/b369006.txt">Table of n, a(n) for n = 1..65537</a>

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

%F a(1) = 0, for n > 1, a(n) = A359550(A083345(n)).

%F A368914(n) <= a(n) <= 1 - A369004(n).

%o (PARI)

%o A083345(n) = { my(f=factor(n)); numerator(vecsum(vector(#f~, i, f[i, 2]/f[i, 1]))); };

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

%o A369006(n) = if(1==n,0,A359550(A083345(n)));

%Y Cf. A003415, A083345, A359550, A368914, A368916 [= a(A276086(n))], A369004, A369007.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 14 2024