%I #13 Jan 06 2024 19:00:50

%S 12,18,20,30,42,48,56,66,70,72,78,80,84,88,90,102,104,108,114,120,138,

%T 150,162,174,180,186,192,196,200,210,220,222,246,252,258,260,264,270,

%U 272,280,282,288,294,300,304,308,312,318,320,330,336,340,354,364,366

%N Numbers k such that Sum_{d|k, d<k, d not occurring before} d > k.

%C This is the complement of the set S occurring in S-perfect numbers A118372.

%C From _Amiram Eldar_, Aug 11 2023: (Start)

%C Sometimes called S-abundant numbers, since they are analogous to abundant numbers (A005101) as S-perfect numbers (A118372) are analogous to perfect numbers (A000396).

%C De Koninck and Ivić conjectured that this sequence has an asymptotic density.

%C The numbers of terms not exceeding 10^k, for k = 2, 3, ..., are 15, 152, 1567, 15336, 154301, 1541445, 15392073, ... . Apparently, the asymptotic density of this sequence exists and equals 0.15... . (End)

%D Elena Deza, Perfect and Amicable Numbers, World Scientific, 2023, pp. 325-327.

%H Donovan Johnson, <a href="/A181487/b181487.txt">Table of n, a(n) for n = 1..10000</a>

%H Jean-Marie De Koninck and Aleksandar Ivić, <a href="https://www.emis.de/journals/PIMB/078/2.html">On a sum of divisors problem</a>, Publications de l'Institut Mathématique (Beograd), New Series, Vol. 64 (78) (1998), pp. 9-20.

%H Gérard Villemin, <a href="http://villemin.gerard.free.fr/aNombre/TYPDIVIS/ParfaitS.htm">Nombres S-PARFAITS ou Nombres de Granville</a>, NOMBRES - Curiosités, théorie et usages, 2019 (in French).

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Granville_number">Granville number</a>.

%t seq[kmax_] := Module[{s = {1}, a = {}, sum}, Do[sum = Total[Select[Divisors[k], MemberQ[s, #] &]]; If[sum <= k, AppendTo[s, k], AppendTo[a, k]], {k, 2, kmax}]; a]; seq[400] (* _Amiram Eldar_, Aug 11 2023 *)

%o (PARI) A181487(Nmax) = { my(C=0); for(n=2,Nmax, sumdiv(n,d,!bittest(C,d)*d)>2*n & !print1(n", ") & C+=1<<n )}

%Y Cf. A000396, A005101, A118372.

%K nonn

%O 1,1

%A _M. F. Hasler_, Oct 28 2010