A071322 Alternating sum of all prime factors of n; primes nonincreasing, starting with the largest prime factor: A006530(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, 10, 67, 17, 20, 4, 71, 2, 73, 35

Offset

1,2

Comments

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

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

Links

Shreyansh Jaiswal, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale).

Example

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

Mathematica

aspf[n_]:=Total[Times@@@Partition[Riffle[Reverse[Flatten[Table[#[[1]], {#[[2]]}]&/@FactorInteger[n]]], {1, -1}, {2, -1, 2}], 2]]; Join[{0}, Array[ aspf, 80, 2]] (* Harvey P. Dale, Apr 19 2015 *)

Prog

(Python)
from sympy import factorint
def c(n):
    fs = factorint(n, multiple=True)
    return sum(fs[::2])-sum(fs[1::2])
for n in range(1, 75):
    print(abs(c(n)), end=", ") #Shreyansh Jaiswal, Apr 16 2025

Crossrefs

Cf. A001414, A071324, a(n) = abs(A071321(n)).

Sequence in context: A335940 A339767 A071321 * A072594 A353051 A272591

Adjacent sequences: A071319 A071320 A071321 * A071323 A071324 A071325

Keyword

nonn

Author

Reinhard Zumkeller, May 18 2002

Status

approved