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A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.

%I A318464 #24 Aug 10 2024 03:54:24
%S A318464 0,1,1,1,1,2,1,1,1,2,1,2,1,2,2,2,1,2,1,2,2,2,1,2,1,2,1,2,1,3,1,1,2,2,
%T A318464 2,2,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,1,2,2,2,2,2,1,3,1,2,2,2,2,3,1,2,
%U A318464 2,3,1,2,1,2,2,2,2,3,1,3,2,2,1,3,2,2,2,2,1,3,2,2,2,2,2,2,1,2,2,2,1,3,1,2,3
%N A318464 Additive with a(p^e) = A007895(e), where A007895(n) gives the number of terms in Zeckendorf representation of n.
%C A318464 From _Amiram Eldar_, Aug 09 2024: (Start)
%C A318464 The number of factors of n of the form p^Fibonacci(k), where p is a prime and k >= 2, when the factorization is uniquely done using the Zeckendorf representation of the exponents in the prime factorization of n.
%C A318464 Equivalently, the number of Zeckendorf-infinitary divisors of n (defined in A318465) that are prime powers (A246655). (End)
%H A318464 Antti Karttunen, <a href="/A318464/b318464.txt">Table of n, a(n) for n = 1..65537</a>
%H A318464 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A318464 a(n) = A007814(A318465(n)).
%F A318464 a(n) = A001222(A318469(n)).
%F A318464 Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B + C), where B is Mertens's constant (A077761) and C = Sum_{k>=2} (A007895(k)-A007895(k-1)) * P(k) = 0.05631817952062180045..., where P(s) is the prime zeta function. - _Amiram Eldar_, Oct 09 2023
%t A318464 z[n_] := Length[DeleteCases[NestWhileList[# - Fibonacci[Floor[Log[Sqrt[5]*# + 3/2]/Log[GoldenRatio]]] &, n, # > 1 &], 0]]; a[n_] := Total[z /@ FactorInteger[n][[;; , 2]]]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, May 15 2023 *)
%o A318464 (PARI)
%o A318464 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A318464 A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
%o A318464 A318464(n) = vecsum(apply(e -> A007895(e),factor(n)[,2]));
%Y A318464 Cf. A007895, A077761, A246655, A318465, A318469.
%K A318464 nonn
%O A318464 1,6
%A A318464 _Antti Karttunen_, Aug 30 2018