A317146 Moebius function in the ranked poset of factorizations of n into factors > 1, evaluated at the minimum (the prime factorization of n).

0, 1, 1, -1, 1, -1, 1, 0, -1, -1, 1, 1, 1, -1, -1, 0, 1, 1, 1, 1, -1, -1, 1, -1, -1, -1, 0, 1, 1, 2, 1, 0, -1, -1, -1, -1, 1, -1, -1, -1, 1, 2, 1, 1, 1, -1, 1, 1, -1, 1, -1, 1, 1, -1, -1, -1, -1, -1, 1, -3, 1, -1, 1, 0, -1, 2, 1, 1, -1, 2, 1, 2, 1, -1, 1, 1

Offset: 1

Author

Gus Wiseman, Jul 22 2018

Keywords

• sign

Comments

If x and y are factorizations of the same integer and it is possible to produce x by further factoring the factors of y, flattening, and sorting, then x <= y.

Examples

			The factorizations of 60 followed by their Moebius values are the following. The second column sums to 0, as required.
  (2*2*3*5) -> -3
   (2*2*15) ->  1
   (2*3*10) ->  2
    (2*5*6) ->  2
     (2*30) -> -1
    (3*4*5) ->  2
     (3*20) -> -1
     (4*15) -> -1
     (5*12) -> -1
     (6*10) -> -1
       (60) ->  1
		

Crossrefs

Cf. A000837, A001055, A007716, A045778, A162247, A275024, A281113, A299202, A317144, A317145.

Formula

Product_{k>=2} 1/(1-a(n)/n^s) = 1+P(s), Re(s)>1, where P(s) is the prime zeta function. - Tian Vlasic, Jan 25 2024