A074726 - OEIS

%I #30 Feb 05 2022 07:09:37

%S 12,18,24,30,36,40,42,48,54,60,72,80,84,90,96,108,120,126,132,140,144,

%T 150,156,160,162,168,180,192,198,200,204,210,216,225,228,234,240,252,

%U 264,270,276,280,288,294,300,306,312,315,320

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

%C 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

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

%H Amiram Eldar, <a href="/A074726/b074726.txt">Table of n, a(n) for n = 1..219</a> (terms 1..156 from T. D. Noe)

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/">Appendix 1. Factorization results</a> links to internal pages.

%H Charles R. Wall, <a href="https://fq.math.ca/Scanned/20-1/advanced20-1.pdf">Problem H-338</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 20, No. 1 (1982), p. 94; <a href="https://www.fq.math.ca/Scanned/21-2/advanced21-2.pdf">Some Abundance</a>, Solution to Problem H-338 by the proposer, ibid., Vol. 21, No. 2 (1983), pp. 159-160.

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

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

%o (PARI) isok(k) = my(f=fibonacci(k)); sigma(f) > 2*f; \\ _Michel Marcus_, Feb 05 2022

%Y Cf. A000045, A063477, A074316.

%K nonn

%O 1,1

%A _Benoit Cloitre_, Sep 04 2002

%E Edited and extended by _Robert G. Wilson v_, Sep 06 2002