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A318465 The number of Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the Zeckendorf expansion (A014417) of each s(i) contains only terms that are in the Zeckendorf expansion of r(i).

%I A318465 #17 Jan 09 2020 18:12:35
%S A318465 1,2,2,2,2,4,2,2,2,4,2,4,2,4,4,4,2,4,2,4,4,4,2,4,2,4,2,4,2,8,2,2,4,4,
%T A318465 4,4,2,4,4,4,2,8,2,4,4,4,2,8,2,4,4,4,2,4,4,4,4,4,2,8,2,4,4,4,4,8,2,4,
%U A318465 4,8,2,4,2,4,4,4,4,8,2,8,4,4,2,8,4,4,4,4,2,8,4,4,4,4,4,4,2,4,4,4,2,8,2,4,8
%N A318465 The number of Zeckendorf-infinitary divisors of n = Product_{i} p(i)^r(i): divisors d = Product_{i} p(i)^s(i), such that the Zeckendorf expansion (A014417) of each s(i) contains only terms that are in the Zeckendorf expansion of r(i).
%C A318465 Zeckendorf-infinitary divisors are analogous to infinitary divisors (A077609) with Zeckendorf expansion instead of binary expansion. - _Amiram Eldar_, Jan 09 2020
%H A318465 Antti Karttunen, <a href="/A318465/b318465.txt">Table of n, a(n) for n = 1..65537</a>
%H A318465 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A318465 Multiplicative with a(p^e) = 2^A007895(e), where A007895(n) gives the number of terms in the Zeckendorf representation of n.
%F A318465 a(n) = 2^A318464(n).
%e A318465 a(16) = 4 since 16 = 2^4 and the Zeckendorf expansion of 4 is 101, i.e., its Zeckendorf representation is a set with 2 terms: {1, 3}. There are 4 possible exponents of 2: 0, 1, 3 and 4, corresponding to the subsets {}, {1}, {3} and {1, 3}. Thus 16 has 4 Zeckendorf-infinitary divisors: 2^0 = 1, 2^1 = 2, 2^3 = 8, and 2^4 = 16.
%t A318465 fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, _?(# == 1 &)]]]; f[p_, e_] := 2^Length@fb[e]; a[1] = 1; a[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n])); Array[a, 100] (* _Amiram Eldar_, Jan 09 2020 after _Robert G. Wilson v_ at A014417 *)
%o A318465 (PARI)
%o A318465 A072649(n) = { my(m); if(n<1, 0, m=0; until(fibonacci(m)>n, m++); m-2); }; \\ From A072649
%o A318465 A007895(n) = { my(s=0); while(n>0, s++; n -= fibonacci(1+A072649(n))); (s); }
%o A318465 A318465(n) = factorback(apply(e -> 2^A007895(e),factor(n)[,2]));
%Y A318465 Cf. A007895, A318464, A318469.
%Y A318465 Cf. A014417, A037445, A038148, A074848, A077609.
%Y A318465 Cf. also A318472, A318474.
%K A318465 nonn,mult
%O A318465 1,2
%A A318465 _Antti Karttunen_, Aug 30 2018
%E A318465 Name edited and interpretation in terms of divisors added by _Amiram Eldar_, Jan 09 2020