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

%I A129066 #22 Jun 28 2025 09:31:59
%S A129066 1,5,25,125,625,3125,15625,75025,78125,375125,390625,1875625,1953125,
%T A129066 9378125,9765625,46890625,48828125,234453125,244140625,332813125,
%U A129066 1172265625,1220703125,1664065625,5628750625,5861328125,6103515625,8320328125,9006076025
%N A129066 Numbers k such that k divides Fibonacci(k) with multiples of 12 excluded.
%C A129066 Set difference of A023172 and 12*A072378.
%C A129066 The sequence is closed under multiplication.
%C A129066 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)).
%C A129066 In particular, this sequence includes all terms of geometric progressions 5^k*Fibonacci(5^m) for integers k >= 0 and m >= 0.
%H A129066 F. Lengyel, <a href="http://employees.oxy.edu/lengyel/papers/FQp_sect112502M.pdf">Divisibility Properties by Multisection</a>, <a href="https://www.fq.math.ca/41-1.html"> Fib. Quart. 41 (1) 72</a>
%e A129066 a(1) = Fibonacci(1) = 1,
%e A129066 a(2) = Fibonacci(5) = 5,
%e A129066 a(3)..a(7) = {5^2, 5^3, 5^4, 5^5, 5^6},
%e A129066 a(8) = 75025 = 5^2*3001 = Fibonacci(5^2),
%e A129066 a(9) = 5^7,
%e A129066 a(10) = 375125 = 5^3*3001 = 5*Fibonacci(5^2),
%e A129066 a(11) = 5^8.
%t A129066 Do[ If[ !IntegerQ[ n/12 ] && IntegerQ[ Fibonacci[n] / n ], Print[n] ], {n,1,5^8} ]
%o A129066 (PARI) is(n)=n%12 && (Mod([0,1;1,1],n)^n*[0;1])[1,1]==0 \\ _Charles R Greathouse IV_, Nov 04 2016
%Y A129066 Prime divisors are given in A171980. Their smallest multiples are given in A171981.
%Y A129066 Cf. A072378, A023172.
%K A129066 nonn
%O A129066 1,2
%A A129066 _Alexander Adamchuk_, May 11 2007
%E A129066 Edited and extended by _Max Alekseyev_, Sep 20 2009
%E A129066 a(1)=1 added by _Zak Seidov_, Nov 01 2009
%E A129066 Edited and extended by _Max Alekseyev_, Jan 20 2010