A181487 Numbers k such that Sum_{d|k, d k.

12, 18, 20, 30, 42, 48, 56, 66, 70, 72, 78, 80, 84, 88, 90, 102, 104, 108, 114, 120, 138, 150, 162, 174, 180, 186, 192, 196, 200, 210, 220, 222, 246, 252, 258, 260, 264, 270, 272, 280, 282, 288, 294, 300, 304, 308, 312, 318, 320, 330, 336, 340, 354, 364, 366

Offset: 1

Author

M. F. Hasler, Oct 28 2010

Keywords

• nonn

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.

Crossrefs

Cf. A000396, A005101, A118372.

Programs

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 *)

PARI

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