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

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, 42, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 110, 111, 113, 114, 115

Offset: 1

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

• nonn

Comments

A008472(a(n)) == 0 modulo A001221(a(n)).

Examples

			42=2*3*7: (2+3+7)/3=4, therefore 42 is a term.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Union of A246655 and A070005.

Positions of 1's in A323172.

The version counting multiplicity is A078175.

The version for prime indices is A326621.

The average of the set of distinct prime factors is A323171/A323172.

The average of the multiset of prime factors is A123528/A123529.

Cf. A001221, A001414, A067629, A112798, A316413, A316428, A326567/A326568, A328966.

Programs

Haskell

a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013

Mathematica

Select[Range[2,200],IntegerQ[Mean[Transpose[FactorInteger[#]][[1]]]]&] (* Harvey P. Dale, Apr 18 2016 *)

PARI

is(n)=my(f=factor(n)[,1]);sum(i=1,#f,f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013

Formula

a(n) << n log n/(log log n)^k for any k. - Charles R Greathouse IV, May 30 2013