A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n).

0, 2, 3, 0, 5, -1, 7, 2, 0, -3, 11, 3, 13, -5, -2, 0, 17, 2, 19, 5, -4, -9, 23, -1, 0, -11, 3, 7, 29, 4, 31, 2, -8, -15, -2, 0, 37, -17, -10, -3, 41, 6, 43, 11, 5, -21, 47, 3, 0, 2, -14, 13, 53, -1, -6, -5, -16, -27, 59, -2, 61, -29, 7, 0, -8

Offset: 1

Author

Reinhard Zumkeller, May 18 2002

Keywords

• sign

Comments

a(n) = 0 iff n square, a(A000290(n)) = 0;

a(n) <= 0 iff A001222(n) is even;

a(n) = n iff n prime, a(A000040(n)) = A000040(n).

a(2n) = -a(n) + 2. - Ralf Stephan

Examples

			72 = 2*2*2*3*3, therefore a(72) = 2 - 2 + 2 - 3 + 3 = 2;
90 = 2*3*3*5, therefore a(90) = 2 - 3 + 3 - 5 = -3.
		

Links

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

Crossrefs

Cf. A001414, A020639, A071322, A071323.

Cf. A008836.

Cf. A027746, A033999.

Programs

Haskell

a071321 1 = 0
a071321 n = sum $ zipWith (*) a033999_list $ a027746_row n
-- Reinhard Zumkeller, Jun 01 2013

Mathematica

Join[{0},Table[Total[Times@@@Partition[Riffle[Flatten[Table[#[[1]],{#[[2]]}]&/@ FactorInteger[n]],{1,-1},{2,-1,2}],2]],{n,2,100}]] (* Harvey P. Dale, Sep 23 2015 *)

Python

from sympy import factorint
def A071321(n):
    fs = factorint(n,multiple=True)
    return sum(fs[::2])-sum(fs[1::2]) # Chai Wah Wu, Aug 23 2021

Formula

a(n) = -A071322(n)*A008836(n). - Franklin T. Adams-Watters, Oct 18 2006