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Number of terms in Zeckendorf expansion needed to write the second Fibonacci based variant of arithmetic derivative of n.
3

%I #5 Oct 29 2019 21:06:09

%S 0,0,1,1,1,1,3,1,2,2,3,1,2,1,3,3,3,1,4,1,2,3,3,1,3,3,2,3,5,1,2,1,3,4,

%T 2,4,1,1,3,2,3,1,4,1,5,5,5,1,3,3,3,3,4,1,5,4,6,4,4,1,3,1,4,5,3,3,4,1,

%U 3,4,3,1,5,1,6,4,5,3,4,1,3,3,6,1,4,3,6,6,6,1,5,3,5,5,4,5,3,1,2,5,3,1,4,1,4,3

%N Number of terms in Zeckendorf expansion needed to write the second Fibonacci based variant of arithmetic derivative of n.

%H Antti Karttunen, <a href="/A328848/b328848.txt">Table of n, a(n) for n = 0..20000</a>

%F a(n) = A007895(A328846(n)).

%o (PARI)

%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 A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }

%o A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649

%o A328848(n) = A007895(A328846(n));

%Y Cf. A000045, A007895, A324905, A328846, A328847.

%K nonn

%O 0,7

%A _Antti Karttunen_, Oct 29 2019