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A071321 Alternating sum of all prime factors of n; primes nondecreasing, starting with the least prime factor: A020639(n). 38
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

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

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(Haskell)

a071321 1 = 0

a071321 n = sum $ zipWith (*) a033999_list $ a027746_row n

-- Reinhard Zumkeller, Jun 01 2013

(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

CROSSREFS

Cf. A001414, A020639, A071322, A071323.

Cf. A008836.

Cf. A027746, A033999.

Sequence in context: A140664 A335940 A339767 * A071322 A072594 A353051

Adjacent sequences:  A071318 A071319 A071320 * A071322 A071323 A071324

KEYWORD

sign

AUTHOR

Reinhard Zumkeller, May 18 2002

STATUS

approved

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Last modified May 20 01:04 EDT 2022. Contains 353847 sequences. (Running on oeis4.)