A174894 - OEIS

A174894

Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

%I #12 Nov 16 2024 16:07:21

%S 2,3,4,5,7,8,9,11,13,15,16,17,19,21,23,25,27,29,31,32,33,35,37,39,41,

%T 42,43,47,49,51,53,55,57,59,61,64,65,67,69,71,73,77,78,79,81,83,85,87,

%U 89,91,93,95,97,101

%N Numbers such that the arithmetic mean of their distinct prime factors and the arithmetic mean of all of their prime factors are both integers.

%C Subsequence of A078174 and A078175.

%C Complement of A176552. [From _Jaroslav Krizek_, Apr 21 2010]

%H Harvey P. Dale, <a href="/A174894/b174894.txt">Table of n, a(n) for n = 1..1000</a>

%e For a(11) = 16: 16 = 2^4; both (2+2+2+2)/4 and 2/1 are integers.

%t mdmaQ[n_]:=With[{fi=FactorInteger[n]},AllTrue[{Mean[Flatten[Table[#[[1]],#[[2]]]&/@fi]],Mean[fi[[;;,1]]]},IntegerQ]]; Select[Range[ 2,110],mdmaQ] (* _Harvey P. Dale_, Nov 16 2024 *)

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Apr 01 2010

%E Definition clarified by _Harvey P. Dale_, Nov 16 2024