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%I #46 Feb 23 2024 07:28:12
%S 5775,5984,7424,11024,21735,21944,26144,27404,39375,43064,49664,56924,
%T 58695,61424,69615,70784,76544,77175,79695,81080,81675,82004,84524,
%U 84644,89775,91664,98175,103455,104895,106784,109395,111824,116655
%N Numbers k such that both k and k+1 are abundant.
%C Numbers k such that both sigma(k) > 2k and sigma(k+1) > 2*(k+1).
%C Numbers k such that both k and k+1 are in A005101.
%C Set difference of sequences A103289 and {2^m-1} for m in A103291.
%C The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 3, 27, 357, 3723, 36640, 365421, 3665799, 36646071, ... . Apparently, the asymptotic density of this sequence exists and equals 0.000366... . - _Amiram Eldar_, Sep 02 2022
%H T. D. Noe, <a href="/A096399/b096399.txt">Table of n, a(n) for n = 1..10000</a>
%H Yong-Gao Chen, Hui Lv, <a href="http://arxiv.org/abs/1603.06176">On consecutive abundant numbers</a>, arXiv:1603.06176 [math.NT], 2016.
%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1935-03.pdf">Note on consecutive abundant numbers</a>, J. London Math. Soc., 10 (1935), 128-131.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_878.htm">Puzzle 878. Consecutive abundant integers</a>, The Prime Puzzles and Problems Connection.
%e sigma(5775) = sigma(3*5*5*7*11) = 11904 > 2*5775.
%e sigma(5776) = sigma(2*2*2*2*19*19) = 11811 > 2*5776.
%t fQ[n_] := DivisorSigma[1, n] > 2 n; Select[ Range@ 117000, fQ[ # ] && fQ[ # + 1] &] (* _Robert G. Wilson v_, Jun 11 2010 *)
%t Select[Partition[Select[Range[120000], DivisorSigma[1, #] > 2 # &], 2, 1], Differences@ # == {1} &][[All, 1]] (* _Michael De Vlieger_, May 20 2017 *)
%o (PARI) for(i=1,1000000,if(sigma(i)>2*i && sigma(i+1)>2*(i+1),print(i))); \\ _Max Alekseyev_, Jan 28 2005
%Y Cf. A005101, A103289, A103291, A023196.
%K nonn
%O 1,1
%A _John L. Drost_, Aug 06 2004
%E Two further terms from _Max Alekseyev_, Jan 28 2005
%E Entry revised by _N. J. A. Sloane_, Dec 03 2006
%E Edited by _T. D. Noe_, Nov 15 2010
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