A328847 Number of terms in Zeckendorf expansion needed to write the first Fibonacci based variant of arithmetic derivative of n.

0, 0, 1, 1, 2, 1, 2, 1, 3, 3, 2, 1, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 1, 3, 3, 3, 4, 4, 1, 3, 1, 4, 3, 5, 3, 3, 1, 5, 4, 3, 1, 4, 1, 3, 3, 4, 1, 3, 4, 3, 3, 6, 1, 5, 3, 4, 5, 4, 1, 3, 1, 5, 3, 4, 4, 2, 1, 5, 6, 2, 1, 3, 1, 4, 3, 5, 5, 5, 1, 3, 5, 5, 1, 3, 7, 5, 4, 3, 1, 3, 5, 5, 5, 4, 6, 4, 1, 3, 4, 5, 1, 7, 1, 6, 5

Offset: 0

Author

Antti Karttunen, Oct 28 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A000045, A007895, A324907, A328845.

Programs

PARI

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]));
A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
A328847(n) = A007895(A328845(n));

Formula

a(n) = A007895(A328845(n)).

a(p) = 1 for all primes p.