A129066 Numbers k such that k divides Fibonacci(k) with multiples of 12 excluded.

1, 5, 25, 125, 625, 3125, 15625, 75025, 78125, 375125, 390625, 1875625, 1953125, 9378125, 9765625, 46890625, 48828125, 234453125, 244140625, 332813125, 1172265625, 1220703125, 1664065625, 5628750625, 5861328125, 6103515625, 8320328125, 9006076025

Offset: 1

Author

Alexander Adamchuk, May 11 2007

Keywords

• nonn

Comments

Set difference of A023172 and 12*A072378.

The sequence is closed under multiplication.

Also, if m is in this sequence (i.e., gcd(F(m),m)=m) then F(m) is in this sequence (since gcd(F(F(m)),F(m)) = F(gcd(F(m),m)) = F(m)).

In particular, this sequence includes all terms of geometric progressions 5^k*Fibonacci(5^m) for integers k >= 0 and m >= 0.

Examples

			a(1) = Fibonacci(1) = 1,
a(2) = Fibonacci(5) = 5,
a(3)..a(7) = {5^2, 5^3, 5^4, 5^5, 5^6},
a(8) = 75025 = 5^2*3001 = Fibonacci(5^2),
a(9) = 5^7,
a(10) = 375125 = 5^3*3001 = 5*Fibonacci(5^2),
a(11) = 5^8.
		

Links

• F. Lengyel, Divisibility Properties by Multisection, Fib. Quart. 41 (1) 72

Crossrefs

Prime divisors are given in A171980. Their smallest multiples are given in A171981.

Cf. A072378, A023172.

Programs

Mathematica

Do[ If[ !IntegerQ[ n/12 ] && IntegerQ[ Fibonacci[n] / n ], Print[n] ], {n,1,5^8} ]

PARI

is(n)=n%12 && (Mod([0,1;1,1],n)^n*[0;1])[1,1]==0 \\ Charles R Greathouse IV, Nov 04 2016

Extensions

Edited and extended by Max Alekseyev, Sep 20 2009

a(1)=1 added by Zak Seidov, Nov 01 2009

Edited and extended by Max Alekseyev, Jan 20 2010