A309840 If n = Sum (2^e_k) then a(n) = Product (Fibonacci(e_k + 3)).

1, 2, 3, 6, 5, 10, 15, 30, 8, 16, 24, 48, 40, 80, 120, 240, 13, 26, 39, 78, 65, 130, 195, 390, 104, 208, 312, 624, 520, 1040, 1560, 3120, 21, 42, 63, 126, 105, 210, 315, 630, 168, 336, 504, 1008, 840, 1680, 2520, 5040, 273, 546, 819, 1638, 1365, 2730, 4095, 8190

Offset: 0

Author

Ilya Gutkovskiy, Aug 19 2019

Keywords

• nonn

Examples

			23 = 2^0 + 2^1 + 2^2 + 2^4 so a(23) = Fibonacci(3) * Fibonacci(4) * Fibonacci(5) * Fibonacci(7) = 390.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384
• Michael De Vlieger, Fan style binary tree showing a(n) for n = 0..2^14, showing primes in red, perfect prime powers in gold, squarefree composites in green, and numbers neither squarefree nor prime powers in blue and magenta, where magenta additionally represents powerful numbers that are not perfect prime powers and bright green represents primorials.

Crossrefs

Cf. A000045, A003266, A019565, A022290, A029930, A121663, A160009.

Programs

Mathematica

nmax = 55; CoefficientList[Series[Product[(1 + Fibonacci[k + 3] x^(2^k)), {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := Fibonacci[Floor[Log[2, n]] + 3] a[n - 2^Floor[Log[2, n]]]; Table[a[n], {n, 0, 55}]

PARI

a(n)={vecprod([fibonacci(k+2) | k<-Vec(select(b->b, Vecrev(digits(n, 2)), 1))])} \\ Andrew Howroyd, Aug 19 2019

Formula

G.f.: Product_{k>=0} (1 + Fibonacci(k + 3) * x^(2^k)).

a(0) = 1; a(n) = Fibonacci(floor(log_2(n)) + 3) * a(n - 2^floor(log_2(n))).

a(2^(k-2)-1) = A003266(k).