A378859 Achilles numbers that are abundant.

72, 108, 200, 288, 392, 432, 500, 648, 800, 864, 968, 972, 1152, 1352, 1372, 1568, 1800, 1944, 2000, 2592, 2700, 3200, 3456, 3528, 3872, 3888, 4000, 4500, 4608, 5000, 5292, 5400, 5408, 5488, 6272, 6912, 7200, 8712, 8748, 9000, 9248, 9800, 10368, 10584, 10800, 10976, 11552, 12168, 12348, 12500, 12800, 13068, 13500, 14112, 15488

Offset: 1

Author

Massimo Kofler, Dec 09 2024

Keywords

• nonn

Comments

33075 is the smallest odd term.

The set of distinct prime factors of a term can be any set P of primes such that Product_{p in P} p/(p-1) > 2. - Robert Israel, Jan 29 2025

Examples

			72=2^3*3^2 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is smaller than the sum of its proper divisors (1+2+3+4+6+8+9+12+18+24+36=123).
108=2^2*3^3 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is smaller than the sum of its proper divisors (1+2+3+4+6+9+12+18+27+36+54=172).
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A005101 and A052486.

Programs

Maple

filter:= proc(n) local F, E, t; F:= ifactors(n)[2]; E:= F[..,2]; min(E)>1 and igcd(op(E))=1 and mul((t[1]^(1+t[2])-1)/(t[1]-1), t = F) > 2*n end proc:
select(filter, [$1..10^5]); # Robert Israel, Jan 28 2025

Mathematica

q[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; AllTrue[e, # > 1 &] && GCD @@ e == 1 && Times @@ ((p - 1/p^e)/(p - 1)) > 2]; Select[Range[16000], q] (* Amiram Eldar, Dec 09 2024 *)