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%I #18 Jul 05 2023 08:23:55
%S 1,6,15,18,22,38,75,93,106,145,253,289,695,959,1467,1703,1820,1821,
%T 2159,3283,3485,3503,3959,4223,4343,4559,5063,5183,6482,6589,7202,
%U 10081,12895,13501,13526,16422,21040,21246,23329,26461,29521,45033,46369,51409,53821
%N Nonprime numbers k whose arithmetic derivative k' (A003415) is a Fibonacci number (A000045).
%C Only nonprime numbers are considered because for prime p, p' = 1 = A000045(1).
%e 1' = 0 = A000045(0), so 1 is a term.
%e 6' = 5 = A000045(5), so 6 is a term.
%e 18' = 21 = A000045(8), so 18 is a term.
%t 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 *)
%o (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))];
%o (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (issquare(k-8)); \\ A000045
%o isok(k) = !isprime(k) && isfib(ad(k)); \\ _Michel Marcus_, Jul 05 2023
%Y Cf. A000045, A003415.
%K nonn
%O 1,2
%A _Marius A. Burtea_, May 03 2023