A374209 - OEIS

A374209

Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 2, 2, 1, 2, 2, 3, 1, 2, 1, 2, 3, 2, 2, 2, 1, 2, 1, 2, 1, 3, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 3, 3, 1, 2, 1, 2, 3

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OFFSET

1,9

COMMENTS

Indices for the first occurrences of k=0..6 are: 1, 2, 9, 63, 693, 7623, 105105.

The claim a(n) <= bigomega(n) is true because A007895(n) is the minimum number of Fibonacci numbers which sum to n, regardless of adjacency or duplication. See Apr 17 2015 comments there.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..129591

Mathworld, Fibonacci Number

Wikipedia, Zeckendorf's theorem

FORMULA

a(n) = A007895(A113177(n)).

a(p) = 1 for all primes p.

a(n) <= A001222(n), see comments.

PROG

(PARI)

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