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

1,2

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.

Links

Table of n, a(n) for n=1..28.

F. Lengyel, Divisibility Properties by Multisection

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Cf. A072378, A023172.

Sequence in context: A291164 A216126 A335506 * A102169 A060391 A000351

Adjacent sequences: A129063 A129064 A129065 * A129067 A129068 A129069

Keyword

nonn

Author

Alexander Adamchuk, May 11 2007

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

Status

approved