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

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

Eric Weisstein's World of Mathematics, 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
A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i, 1])));
A374209(n) = if(isprime(n), 1, A007895(A113177(n)));

Crossrefs

Cf. A000045, A001222, A007895, A030426, A113177.

Cf. also A328847, A328848.

Sequence in context: A218775 A370558 A191971 * A156051 A091267 A003643

Adjacent sequences: A374206 A374207 A374208 * A374210 A374211 A374212

Keyword

nonn

Author

Antti Karttunen, Jul 02 2024

Status

approved