1,2

a(n) includes all terms of A023172(n) with terms 12*A072378(n) excluded.

No other terms below 10^8.

Since 5*F(m) divides F(5*m), if m is in this sequence then 5*m is. 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, a(n) includes all terms of geometric progressions 5^k*Fibonacci(5^m) for integers k>=0 and m>=0.

Open question: does a(n) include the powers of Fibonacci(5^k) and the terms of the form Fibonacci(5^k)*Fibonacci(5^m); or is it closed under multiplication?

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

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

Cf. A072378, A023172.

Sequence in context: A057831 A014946 A132839 this_sequence A102169 A060391 A000351

Adjacent sequences: A129063 A129064 A129065 this_sequence A129067 A129068 A129069

hard,more,nonn,new

Alexander Adamchuk (alex(AT)kolmogorov.com), May 11 2007

Edited and extended by Max Alekseyev (maxale(AT)gmail.com), Sep 20 2009

a(1)=1 added by Zak Seidov (zakseidov(AT)yahoo.com), Nov 01 2009