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A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).

%I A384913 #7 Jun 12 2025 10:24:42
%S A384913 1,1,1,2,1,1,1,3,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,3,2,1,3,2,1,1,1,6,1,1,
%T A384913 1,4,1,1,1,3,1,1,1,2,2,1,1,4,2,2,1,2,1,3,1,3,1,1,1,2,1,1,2,8,1,1,1,2,
%U A384913 1,1,1,6,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A384913 The number of unordered factorizations of n into exponentially Fibonacci powers of primes (A115975).
%C A384913 First differs from A384912 at n = 64.
%H A384913 Amiram Eldar, <a href="/A384913/b384913.txt">Table of n, a(n) for n = 1..10000</a>
%F A384913 Multiplicative with a(p^e) = A003107(e).
%F A384913 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 2.05893526314055968638..., where f(x) = (1-x) / Product_{k>=2} (1-x^A000045(k)).
%e A384913 a(4) = 2 since 4 has 2 factorizations: 2^1 * 2^1 and 2^2, with exponents 1 and 2 that are Fibonacci numbers.
%t A384913 fib[n_] := Boole[Or @@ IntegerQ /@ Sqrt[5*n^2 + {-4, 4}]];
%t A384913 s[n_] := s[n] = If[n == 0, 1, Sum[Sum[d * fib[d], {d, Divisors[j]}] * s[n-j], {j, 1, n}] / n];
%t A384913 f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384913 (PARI) isfib(n) = issquare(5*n^2 - 4) || issquare(5*n^2 + 4);
%o A384913 s(n) = if(n < 1, 1, sum(j = 1, n, sumdiv(j, d, d*isfib(d)) * s(n-j))/n);
%o A384913 a(n) = vecprod(apply(s, factor(n)[, 2]));
%Y A384913 Cf. A000045, A003107, A115063, A115975.
%Y A384913 Similar sequences: A000688, A046951, A050361, A050377, A188581, A188585, A322885, A370256, A384912, A384914, A384915, A384916.
%K A384913 nonn,easy,mult
%O A384913 1,4
%A A384913 _Amiram Eldar_, Jun 12 2025