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

1,1

Comments

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(PARI) is(n)=my(f=factor(n)[, 1]); sum(i=1, #f, f[i])%#f==0 \\ Charles R Greathouse IV, May 30 2013
(Haskell)
a078174 n = a078174_list !! (n-1)
a078174_list = filter (\x -> a008472 x `mod` a001221 x == 0) [2..]
-- Reinhard Zumkeller, Jun 01 2013

Crossrefs

Union of A000961 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.

Sequence in context: A078175 A213715 A360552 * A174894 A275616 A088948

Adjacent sequences: A078171 A078172 A078173 * A078175 A078176 A078177

Keyword

nonn

Author

Reinhard Zumkeller, Nov 20 2002

Status

approved