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A078175 Numbers with an integer arithmetic mean of all prime factors.

%I A078175 #16 Jun 23 2021 18:11:17
%S A078175 2,3,4,5,7,8,9,11,13,15,16,17,19,20,21,23,25,27,29,31,32,33,35,37,39,
%T A078175 41,42,43,44,47,49,50,51,53,55,57,59,60,61,64,65,67,68,69,71,73,77,78,
%U A078175 79,81,83,85,87,89,91,92,93,95,97,101,103,105,107,109,110,111,112,113
%N A078175 Numbers with an integer arithmetic mean of all prime factors.
%C A078175 A001414(a(n)) == 0 modulo A001222(a(n)).
%H A078175 Reinhard Zumkeller, <a href="/A078175/b078175.txt">Table of n, a(n) for n = 1..10000</a>
%e A078175 2100=2*2*3*5*5*7: (2+2+3+5+5+7)/6=4, therefore 2100 is a term.
%t A078175 sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
%t A078175 filterQ[n_] := Divisible[sopfr[n], PrimeOmega[n]];
%t A078175 Select[Range[2, 1000], filterQ] (* _Jean-François Alcover_, Apr 06 2021 *)
%t A078175 iamQ[n_]:=IntegerQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]]]; Select[Range[2,150],iamQ] (* _Harvey P. Dale_, Jun 23 2021 *)
%o A078175 (Haskell)
%o A078175 a078175 n = a078175_list !! (n-1)
%o A078175 a078175_list = filter (\x -> (a001414 x) `mod` (a001222 x) == 0) [2..]
%o A078175 -- _Reinhard Zumkeller_, Nov 20 2011
%Y A078175 Cf. A001222, A001414, A078174.
%Y A078175 Subsequences: A000040, A000079, A200612.
%K A078175 nonn
%O A078175 1,1
%A A078175 _Reinhard Zumkeller_, Nov 20 2002