cp's OEIS Frontend

A078174 Numbers with an integer arithmetic mean of distinct prime factors.

%I A078174 #22 Jun 19 2025 00:23:28
%S A078174 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 A078174 42,43,45,47,49,51,53,55,57,59,61,63,64,65,67,69,71,73,75,77,78,79,81,
%U A078174 83,84,85,87,89,91,93,95,97,99,101,103,105,107,109,110,111,113,114,115
%N A078174 Numbers with an integer arithmetic mean of distinct prime factors.
%C A078174 A008472(a(n)) == 0 modulo A001221(a(n)).
%H A078174 Reinhard Zumkeller, <a href="/A078174/b078174.txt">Table of n, a(n) for n = 1..10000</a>
%F A078174 a(n) << n log n/(log log n)^k for any k. - _Charles R Greathouse IV_, May 30 2013
%e A078174 42=2*3*7: (2+3+7)/3=4, therefore 42 is a term.
%t A078174 Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* _Harvey P. Dale_, Apr 18 2016 *)
%o A078174 (PARI) is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ _Charles R Greathouse IV_, May 30 2013
%o A078174 (Haskell)
%o A078174 a078174 n = a078174_list !! (n-1)
%o A078174 a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
%o A078174 -- _Reinhard Zumkeller_, Jun 01 2013
%Y A078174 Union of A246655 and A070005.
%Y A078174 Positions of 1's in A323172.
%Y A078174 The version counting multiplicity is A078175.
%Y A078174 The version for prime indices is A326621.
%Y A078174 The average of the set of distinct prime factors is A323171/A323172.
%Y A078174 The average of the multiset of prime factors is A123528/A123529.
%Y A078174 Cf. A001221, A001414, A067629, A112798, A316413, A316428, A326567/A326568, A328966.
%K A078174 nonn
%O A078174 1,1
%A A078174 _Reinhard Zumkeller_, Nov 20 2002