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A374209 Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).

%I A374209 #16 Feb 19 2025 11:57:29
%S A374209 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,
%T A374209 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,
%U A374209 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
%N A374209 Number of terms in Zeckendorf representation needed to write A113177(n), where A113177 is fully additive with a(p) = Fibonacci(p).
%C A374209 Indices for the first occurrences of k=0..6 are: 1, 2, 9, 63, 693, 7623, 105105.
%C A374209 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.
%H A374209 Antti Karttunen, <a href="/A374209/b374209.txt">Table of n, a(n) for n = 1..129591</a>
%H A374209 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A374209 Wikipedia, <a href="https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem">Zeckendorf's theorem</a>.
%F A374209 a(n) = A007895(A113177(n)).
%F A374209 a(p) = 1 for all primes p.
%F A374209 a(n) <= A001222(n), see comments.
%o A374209 (PARI)
%o A374209 A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
%o A374209 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A374209 A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
%o A374209 A374209(n) = if(isprime(n), 1, A007895(A113177(n)));
%Y A374209 Cf. A000045, A001222, A007895, A030426, A113177.
%Y A374209 Cf. also A328847, A328848.
%K A374209 nonn
%O A374209 1,9
%A A374209 _Antti Karttunen_, Jul 02 2024