A362141 Nonprime numbers k whose arithmetic derivative k' (A003415) is a Fibonacci number (A000045).

1, 6, 15, 18, 22, 38, 75, 93, 106, 145, 253, 289, 695, 959, 1467, 1703, 1820, 1821, 2159, 3283, 3485, 3503, 3959, 4223, 4343, 4559, 5063, 5183, 6482, 6589, 7202, 10081, 12895, 13501, 13526, 16422, 21040, 21246, 23329, 26461, 29521, 45033, 46369, 51409, 53821

Offset

1,2

Comments

Only nonprime numbers are considered because for prime p, p' = 1 = A000045(1).

Links

Table of n, a(n) for n=1..45.

Example

1' = 0 = A000045(0), so 1 is a term.
6' = 5 = A000045(5), so 6 is a term.
18' = 21 = A000045(8), so 18 is a term.

Mathematica

fibQ[n_] := Or @@ IntegerQ /@ Sqrt[5 n^2 + {-4, 4}]; d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Range[54000], ! PrimeQ[#] && fibQ[d[#]] &] (* Amiram Eldar, May 05 2023 *)

Prog

(Magma) f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [p:p in [1..54000]|not IsPrime(p) and (IsSquare(5*u*u-4) or IsSquare(5*u*u+4)) where u is Floor(f(p))];
(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (issquare(k-8)); \\ A000045
isok(k) = !isprime(k) && isfib(ad(k)); \\ Michel Marcus, Jul 05 2023

Crossrefs

Cf. A000045, A003415.

Sequence in context: A128693 A105285 A138922 * A044058 A105139 A002599

Adjacent sequences: A362138 A362139 A362140 * A362142 A362143 A362144

Keyword

nonn

Author

Marius A. Burtea, May 03 2023

Status

approved