A067629 The average of the prime factors of n, rounded off to the nearest integer (rounding up if there's a choice), with each factor weighted according to its frequency of occurrence in the prime factorization.

2, 3, 2, 5, 3, 7, 2, 3, 4, 11, 2, 13, 5, 4, 2, 17, 3, 19, 3, 5, 7, 23, 2, 5, 8, 3, 4, 29, 3, 31, 2, 7, 10, 6, 3, 37, 11, 8, 3, 41, 4, 43, 5, 4, 13, 47, 2, 7, 4, 10, 6, 53, 3, 8, 3, 11, 16, 59, 3, 61, 17, 4, 2, 9, 5, 67, 7, 13, 5, 71, 2, 73, 20, 4, 8, 9, 6, 79, 3, 3, 22, 83, 4, 11, 23, 16, 4

Offset: 2

Author

Joseph L. Pe, Feb 02 2002

Keywords

• easy
• nonn

Examples

			24 = 2^3 * 3^1, so the average of the prime factors = (2 + 2 + 2 + 3)/4, which rounded = 2. So a(24) = 2.
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A078175, A126594.

Programs

Maple

with(numtheory): for n from 2 to 150 do printf(`%d,`,round(sum(ifactors(n)[2][i][1]*ifactors(n)[2][i][2], i=1..nops(ifactors(n)[2]))/sum(ifactors(n)[2][i][2], i=1..nops(ifactors(n)[2]) ) )) od:

Mathematica

a[n_] := Floor[1/2+(Plus@@(Times@@#&/@(fn=FactorInteger[n])))/(Plus@@Last/@fn)]

Extensions

Edited by Dean Hickerson and James Sellers, Feb 12 2002