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

1,1

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

Links

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

Example

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).

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 *)

Crossrefs

Intersection of A005101 and A052486.

Sequence in context: A272191 A072412 A052486 * A114128 A375074 A375143

Adjacent sequences: A378854 A378856 A378858 * A378860 A378861 A378862

Keyword

nonn

Author

Massimo Kofler, Dec 09 2024

Status

approved