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A337448 The numbers k for which Fibonacci(k) are Niven numbers.

%I A337448 #23 Sep 08 2022 08:46:25
%S A337448 1,2,3,4,5,6,8,12,18,36,54,72,84,112,120,144,160,180,198,200,216,240,
%T A337448 243,264,286,288,299,324,358,360,468,504,528,540,576,648,720,780,816,
%U A337448 1008,1020,1044,1200,1248,1260,1500,1602,1824,1872,1917,2160,2184,2760
%N A337448 The numbers k for which Fibonacci(k) are Niven numbers.
%C A337448 For a(7) = 8, Fibonacci(8) = 21 and 21/digsum(21) = 7 is a prime number, so Fibonacci(8) is a Moran number (A001101). It seems that this is the only Moran number among the first 100000 Fibonacci numbers.
%H A337448 Chai Wah Wu, <a href="/A337448/b337448.txt">Table of n, a(n) for n = 1..1000</a>
%e A337448 Fibonacci(1) = 1 = A005349(1), so 1 is a term.
%e A337448 Fibonacci(8) = 21 = A005349(14), so 8 is a term.
%e A337448 Fibonacci(12) = 144 = A005349(8), so 12 is a term.
%e A337448 Fibonacci(18) = 2584 = A005349(514), so 18 is a term.
%t A337448 nivenQ[n_] := Divisible[n, Plus @@ IntegerDigits[n]]; Select[Range[3000], nivenQ[Fibonacci[#]] &] (* _Amiram Eldar_, Sep 15 2020 *)
%o A337448 (Magma) niven:=func<n|n mod &+Intseq(n) eq 0>; [k:k in [1..2760]| niven(Fibonacci(k))];
%o A337448 (PARI) isok(k) = my(f=fibonacci(k)); (f % sumdigits(f)) == 0; \\ _Michel Marcus_, Sep 15 2020
%Y A337448 Cf. A000045, A004090, A001101, A005349, A117774, A337449.
%K A337448 nonn,base
%O A337448 1,2
%A A337448 _Marius A. Burtea_, Sep 14 2020