A007436 Moebius transform of Fibonacci numbers. (Formerly M1023)

1, 0, 1, 2, 4, 6, 12, 18, 32, 50, 88, 134, 232, 364, 604, 966, 1596, 2544, 4180, 6708, 10932, 17622, 28656, 46206, 75020, 121160, 196384, 317432, 514228, 831374, 1346268, 2177322, 3524488, 5701290, 9227448, 14927632, 24157816, 39083988, 63245752, 102327372

Offset

1,4

Comments

After a(4) = 2, there are no primes in this sequence. Every element thereafter has at least two prime factors, the semiprimes (intersection of A007436 and A001358) starting 4, 6, 134, 831374, ... - Jonathan Vos Post, Dec 15 2004

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..4786 (first 500 terms from T. D. Noe)

Marta Na Chen and Wenchang Chu, Generating functions for circular sums of binomial products, AIMS Math. 10(12), 2025, 28182-28206. See p. 28198.

N. J. A. Sloane, Transforms.

Formula

Row sums of the triangle generated by A054525 * A127647. - Gary W. Adamson, Jan 22 2007

G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = x/(1 - x - x^2). - Ilya Gutkovskiy, Apr 25 2017

Maple

a:= proc(n) option remember; combinat[fibonacci](n)
      -add(a(d), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..42);  # Alois P. Heinz, Oct 04 2025

Mathematica

mt[n_] := Block[{d = Divisors[n]}, Plus @@ (MoebiusMu /@ (n/d)*Fibonacci /@ d)]; Table[ mt[n], {n, 38}] (* Robert G. Wilson v Dec 10 2004 *)
a[n_] := DivisorSum[n, Fibonacci[#] MoebiusMu[n/#]&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)

Prog

(PARI) a(n)=sumdiv(n, d, fibonacci(d)*moebius(n/d))

Crossrefs

Cf. A000045, A001358, A008683.

Sequence in context: A192224 A167777 A259941 * A052847 A331933 A052823

Adjacent sequences: A007433 A007434 A007435 * A007437 A007438 A007439

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Robert G. Wilson v, Dec 10 2004

Status

approved