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%I #6 Dec 22 2024 16:55:43
%S 675,1125,1323,2312,2888,3087,3267,4232,4563,5324,6075,6125,6728,7688,
%T 7803,8575,8788,9747,10125,10952,11907,11979,13448,14283,14792,15125,
%U 16875,17672,19652,19773,21125,22472,22707,25947,27436,27783,27848,28125,29403,29768,30375,33275,35912,36125,36963,40328
%N Achilles numbers that are deficient.
%e 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).
%e 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).
%t 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 *)
%Y Intersection of A005100 and A052486.
%Y Cf. A378859.
%K nonn
%O 1,1
%A _Massimo Kofler_, Dec 17 2024