

A129066


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


3



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
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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



