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A289586 Numbers k whose smallest multiple that is a Fibonacci number is Fibonacci(k).

%I A289586 #32 Aug 11 2017 06:47:30
%S A289586 1,5,12,25,60,125,300,625,1500,3125,7500,15625,37500,78125,187500,
%T A289586 390625,937500,1953125,4687500,9765625,23437500,48828125,117187500,
%U A289586 244140625,585937500,1220703125,2929687500,6103515625,14648437500,30517578125,73242187500,152587890625,366210937500
%N A289586 Numbers k whose smallest multiple that is a Fibonacci number is Fibonacci(k).
%C A289586 Alternative names:
%C A289586 Numbers k such that Fibonacci(k) is the smallest positive Fibonacci number that is divisible by k.
%C A289586 Numbers that are their own Fibonacci entry points.
%C A289586 Numbers k such that k = A001177(k).
%C A289586 Numbers that are either a power of 5 or 12 times a power of 5. - _Robert Israel_, Aug 07 2017
%H A289586 Robert Israel, <a href="/A289586/b289586.txt">Table of n, a(n) for n = 1..2858</a>
%H A289586 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5).
%F A289586 From _Robert Israel_, Aug 07 2017: (Start)
%F A289586 a(2*k) = 5^k for k >= 1.
%F A289586 a(2*k-1) = 12*5^(k-2) for k >= 2.
%F A289586 G.f.: (1+5*x+7*x^2)/(1-5*x^2). (End)
%e A289586 Fibonacci(25) = 75025 = 25*3001 is the smallest Fibonacci number that is divisible by 25, so 25 is in the sequence.
%e A289586 Although Fibonacci(24) = 46368 = 24*1932 is divisible by 24, it is not the smallest Fibonacci number that is divisible by 24, so 24 is not in the sequence.
%p A289586 1,seq(op([5^k,12*5^(k-1)]), k=1..100); # _Robert Israel_, Aug 07 2017
%Y A289586 Subsequence of A023172 ("Self-Fibonacci numbers").
%Y A289586 Cf. A000045, A001177, A000351 (bisection), A216491 (bisection)
%Y A289586 (Cf. A001602 for a different definition of "Fibonacci entry point".)
%K A289586 nonn,easy
%O A289586 1,2
%A A289586 _Jon E. Schoenfield_, Aug 06 2017
%E A289586 More terms from _Robert Israel_, Aug 07 2017