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a(n) = 1 if there is no prime p such that p^p divides A342001(n), but for A003415(n) such a prime exists, otherwise 0. Here A003415(n) is the arithmetic derivative of n, and A342001(n) = A003415(n) / A003557(n).
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%I #18 Jan 14 2024 17:28:32

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

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

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

%N a(n) = 1 if there is no prime p such that p^p divides A342001(n), but for A003415(n) such a prime exists, otherwise 0. Here A003415(n) is the arithmetic derivative of n, and A342001(n) = A003415(n) / A003557(n).

%C Question: What is the asymptotic mean of this sequence?

%H Antti Karttunen, <a href="/A368913/b368913.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(n) = A368914(n) - A368915(n).

%F For all n >= 0, a(A048103(n)) = a(A276086(n)) = 0.

%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 A003557(n) = (n/factorback(factorint(n)[, 1]));

%o A342001(n) = (A003415(n) / A003557(n));

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

%o A368914(n) = ((n>1)&&A359550(A342001(n)));

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

%o A368913(n) = (A368914(n)-A368915(n));

%Y Characteristic function of A368903.

%Y Cf. A003415, A048103, A276086, A342001, A359550, A368914, A368915.

%K nonn

%O 1

%A _Antti Karttunen_, Jan 09 2024