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a(n) = 1 if A328845(n) and A328846(n) are both even, otherwise 0, where A328845 and A328846 are the first and second Fibonacci-based variants of the arithmetic derivative.
4

%I #7 Jun 28 2024 14:00:17

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

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

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

%N a(n) = 1 if A328845(n) and A328846(n) are both even, otherwise 0, where A328845 and A328846 are the first and second Fibonacci-based variants of the arithmetic derivative.

%H Antti Karttunen, <a href="/A374036/b374036.txt">Table of n, a(n) for n = 0..100000</a>

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

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>

%F a(n) = A374045(n) * A374048(n).

%F a(n) = A059841(A374035(n)).

%o (PARI)

%o A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));

%o A328846(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(2+primepi(f[i,1]))/f[i, 1]));

%o A374036(n) = (!(A328845(n)%2) && !(A328846(n)%2));

%Y Characteristic function of A374037, whose complement A374038 gives the indices of 0's.

%Y Cf. A000045, A000720, A059841, A328845, A328846, A374035, A374045, A374048.

%K nonn

%O 0

%A _Antti Karttunen_, Jun 28 2024