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A362141
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Nonprime numbers k whose arithmetic derivative k' (A003415) is a Fibonacci number (A000045).
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1
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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
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OFFSET
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1,2
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COMMENTS
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Only nonprime numbers are considered because for prime p, p' = 1 = A000045(1).
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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