A124105 - OEIS

%I #13 Feb 16 2025 08:33:03

%S 1,2,8,144,46368,832040,14930352,267914296,4807526976,1548008755920,

%T 498454011879264,160500643816367088,2880067194370816120,

%U 51680708854858323072,16641027750620563662096,5358359254990966640871840

%N Practical Fibonacci numbers.

%C Melfi proves that this sequence is infinite. The first few practical Fibonacci numbers have indices that are themselves practical (analogous to the property that the prime Fibonacci numbers have prime indices) but Melfi observes that this property is not true in general: F444 is practical although 444 itself is not.

%C The indices of these Fibonacci numbers are 1 (and 2), 3, 6, 12, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 126, 132, 144, 150, 156, 168, 180, 192, 204, 210, 216, 228, 240, 252, 264, 270, 276, 288, 294, 300, 312, 324, 330, 336, 348, 360, 378, 384, 390, 396, 408, 420, 432, 444, ... - _Amiram Eldar_, May 29 2017

%H Charles R Greathouse IV, <a href="/A124105/b124105.txt">Table of n, a(n) for n = 1..108</a>

%H Giuseppe Melfi, <a href="http://www.dm.unipi.it/gauss-pages/melfi/public_html/articoli/smapoto.ps">A survey on practical numbers</a>, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.

%H Eric W. Weisstein, From MathWorld: <a href="https://mathworld.wolfram.com/PracticalNumber.html">Practical Number.</a>

%e 144 is a member of this sequence because it is the 12th Fibonacci number and is also a practical number.

%o (PARI) is_A005153(n)=if(n%2,return(n==1)); my(P=1,f=factor(n)); for(i=2, #f~, if(f[i,1]>1+(P*=sigma(f[i-1,1]^f[i-1,2])), return(0))); n>0

%o print1(1); forstep(n=3,200,3,if(is_A005153(t=fibonacci(n)), print1(", "t))) \\ _Charles R Greathouse IV_, Oct 06 2013

%Y Cf. A000045, A005153, A074316.

%K nonn

%O 1,2

%A _David Eppstein_, Dec 13 2006

%E More terms from _Charles R Greathouse IV_, Oct 06 2013