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

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, 41, 42, 43, 44, 47, 49, 50, 51, 53, 55, 57, 59, 60, 61, 64, 65, 67, 68, 69, 71, 73, 77, 78, 79, 81, 83, 85, 87, 89, 91, 92, 93, 95, 97, 101, 103, 105, 107, 109, 110, 111, 112, 113

Offset: 1

Author

Reinhard Zumkeller, Nov 20 2002

Keywords

• nonn

Comments

A001414(a(n)) == 0 modulo A001222(a(n)).

Examples

			2100=2*2*3*5*5*7: (2+2+3+5+5+7)/6=4, therefore 2100 is a term.
		

Links

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

Crossrefs

Cf. A001222, A001414, A078174.

Subsequences: A000040, A000079, A200612.

Programs

Haskell

a078175 n = a078175_list !! (n-1)
a078175_list = filter (\x -> (a001414 x) `mod` (a001222 x) == 0) [2..]
-- Reinhard Zumkeller, Nov 20 2011

Mathematica

sopfr[n_] := If[n == 1, 0, Total[Times @@@ FactorInteger[n]]];
filterQ[n_] := Divisible[sopfr[n], PrimeOmega[n]];
Select[Range[2, 1000], filterQ] (* Jean-François Alcover, Apr 06 2021 *)
iamQ[n_]:=IntegerQ[Mean[Flatten[Table[#[[1]],#[[2]]]&/@ FactorInteger[ n]]]]; Select[Range[2,150],iamQ] (* Harvey P. Dale, Jun 23 2021 *)