OFFSET
1,1
COMMENTS
This is the complement of the set S occurring in S-perfect numbers A118372.
From Amiram Eldar, Aug 11 2023: (Start)
Sometimes called S-abundant numbers, since they are analogous to abundant numbers (A005101) as S-perfect numbers (A118372) are analogous to perfect numbers (A000396).
De Koninck and Ivić conjectured that this sequence has an asymptotic density.
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)
REFERENCES
Elena Deza, Perfect and Amicable Numbers, World Scientific, 2023, pp. 325-327.
LINKS
Donovan Johnson, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck and Aleksandar Ivić, On a sum of divisors problem, Publications de l'Institut Mathématique (Beograd), New Series, Vol. 64 (78) (1998), pp. 9-20.
Gérard Villemin, Nombres S-PARFAITS ou Nombres de Granville, NOMBRES - Curiosités, théorie et usages, 2019 (in French).
Wikipedia, Granville number.
MATHEMATICA
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 *)
PROG
(PARI) A181487(Nmax) = { my(C=0); for(n=2, Nmax, sumdiv(n, d, !bittest(C, d)*d)>2*n & !print1(n", ") & C+=1<<n )}
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Oct 28 2010
STATUS
approved