A379164 Achilles numbers that are deficient.
675, 1125, 1323, 2312, 2888, 3087, 3267, 4232, 4563, 5324, 6075, 6125, 6728, 7688, 7803, 8575, 8788, 9747, 10125, 10952, 11907, 11979, 13448, 14283, 14792, 15125, 16875, 17672, 19652, 19773, 21125, 22472, 22707, 25947, 27436, 27783, 27848, 28125, 29403, 29768, 30375, 33275, 35912, 36125, 36963, 40328
Offset: 1
Keywords
Examples
675=3^3*5^2 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is larger than the sum of its proper divisors (1+3+5+9+15+25+27+45+75+135+225=565). 1125=3^2*5^3 is a term because it is an Achilles number (powerful but imperfect, see A052486) and it is larger than the sum of its proper divisors (1+3+5+9+15+25+45+75+125+225+ 375=903).
Programs
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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[42000], q] (* Amiram Eldar, Dec 17 2024 *)