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

1, 5, 12, 25, 60, 125, 300, 625, 1500, 3125, 7500, 15625, 37500, 78125, 187500, 390625, 937500, 1953125, 4687500, 9765625, 23437500, 48828125, 117187500, 244140625, 585937500, 1220703125, 2929687500, 6103515625, 14648437500, 30517578125, 73242187500, 152587890625, 366210937500

Offset: 1

Author

Jon E. Schoenfield, Aug 06 2017

Keywords

• nonn
• easy

Comments

Alternative names:

Numbers k such that Fibonacci(k) is the smallest positive Fibonacci number that is divisible by k.

Numbers that are their own Fibonacci entry points.

Numbers k such that k = A001177(k).

Numbers that are either a power of 5 or 12 times a power of 5. - Robert Israel, Aug 07 2017

Examples

			Fibonacci(25) = 75025 = 25*3001 is the smallest Fibonacci number that is divisible by 25, so 25 is in the sequence.
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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2858
• Index entries for linear recurrences with constant coefficients, signature (0,5).

Crossrefs

Subsequence of A023172 ("Self-Fibonacci numbers").

Cf. A000045, A001177, A000351 (bisection), A216491 (bisection)

(Cf. A001602 for a different definition of "Fibonacci entry point".)

Programs

Maple

1,seq(op([5^k,12*5^(k-1)]), k=1..100); # Robert Israel, Aug 07 2017

Formula

From Robert Israel, Aug 07 2017: (Start)

a(2*k) = 5^k for k >= 1.

a(2*k-1) = 12*5^(k-2) for k >= 2.

G.f.: (1+5*x+7*x^2)/(1-5*x^2). (End)

Extensions

More terms from Robert Israel, Aug 07 2017