A074316 -id:A074316 - cp's OEIS frontend

A124105 Practical Fibonacci numbers.

1, 2, 8, 144, 46368, 832040, 14930352, 267914296, 4807526976, 1548008755920, 498454011879264, 160500643816367088, 2880067194370816120, 51680708854858323072, 16641027750620563662096, 5358359254990966640871840, 96151855463018422468774568, 1725375039079340637797070384

Offset: 1

David Eppstein, Dec 13 2006

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.
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
Observation: at least the first 54 terms coincide with the 2-dense Fibonacci numbers (the intersection of A000045 and A174973: i.e., those Fibonacci numbers whose divisors increase by a factor of at most 2). Is this the same? If not, where is the first place these sequences differ? - Omar E. Pol, Aug 29 2025

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

Charles R Greathouse IV, Table of n, a(n) for n = 1..108
Giuseppe Melfi, A survey on practical numbers, Rend. Sem. Mat. Univ. Pol. Torino, 53, (1995), 347-359.
Eric Weisstein's World of Mathematics, Practical Number.

Intersection of A000045 and A005153.

Cf. A074316.

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
print1(1); forstep(n=3,200,3,if(is_A005153(t=fibonacci(n)), print1(", "t))) \\ Charles R Greathouse IV, Oct 06 2013

More terms from Charles R Greathouse IV, Oct 06 2013

A074726 Numbers k such that sigma(F(k)) > 2*F(k) where F(k) is the k-th Fibonacci number.

Original entry on oeis.org

12, 18, 24, 30, 36, 40, 42, 48, 54, 60, 72, 80, 84, 90, 96, 108, 120, 126, 132, 140, 144, 150, 156, 160, 162, 168, 180, 192, 198, 200, 204, 210, 216, 225, 228, 234, 240, 252, 264, 270, 276, 280, 288, 294, 300, 306, 312, 315, 320

Offset: 1

Benoit Cloitre, Sep 04 2002

nonn

Conjecture: sigma(F(n)) > 2*F(n) if and only if F(n) is a Zumkeller number except for n = 12. Verified for n <= 371. - M. Farrokhi D. G., Aug 16 2020

The asymptotic density of this sequence is larger than 184/1225 = 0.1502... (Wall, 1982). - Amiram Eldar, Feb 05 2022

Amiram Eldar, Table of n, a(n) for n = 1..219 (terms 1..156 from T. D. Noe)
Hisanori Mishima, Appendix 1. Factorization results links to internal pages.
Charles R. Wall, Problem H-338, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. 1 (1982), p. 94; Some Abundance, Solution to Problem H-338 by the proposer, ibid., Vol. 21, No. 2 (1983), pp. 159-160.

Cf. A000045, A063477, A074316.

Select[ Range[256], DivisorSigma[1, Fibonacci[ #1]] > 2*Fibonacci[ #1] & ]

isok(k) = my(f=fibonacci(k)); sigma(f) > 2*f; \\ Michel Marcus, Feb 05 2022

It seems that a(n) is asymptotic to c*n with 6 < c < 6.5.

Edited and extended by Robert G. Wilson v, Sep 06 2002

cp's OEIS Frontend

A124105 Practical Fibonacci numbers.

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