cp's OEIS Frontend

A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

%I A339564 #5 Apr 19 2021 15:00:43
%S A339564 0,1,1,2,1,3,1,4,2,3,1,7,1,3,3,7,1,7,1,7,3,3,1,14,2,3,4,7,1,10,1,12,3,
%T A339564 3,3,17,1,3,3,14,1,10,1,7,7,3,1,26,2,7,3,7,1,14,3,14,3,3,1,25,1,3,7,
%U A339564 19,3,10,1,7,3,10,1,36,1,3,7,7,3,10,1,26,7,3
%N A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).
%F A339564 a(n) = A057567(n) - A001055(n).
%F A339564 a(n) = Sum_{d|n, d>1} A001055(n/d).
%e A339564 The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
%e A339564   ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
%e A339564          ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
%e A339564                   (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
%e A339564                            ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
%e A339564                                       (3*(4))    (4*(6))      (2*(15))
%e A339564                                       ((2)*2*3)  ((2)*12)     ((3)*10)
%e A339564                                       (2*2*(3))  (2*(12))     (3*(10))
%e A339564                                                  ((2)*2*6)    ((2)*3*5)
%e A339564                                                  (2*2*(6))    (2*(3)*5)
%e A339564                                                  ((2)*3*4)    (2*3*(5))
%e A339564                                                  (2*(3)*4)
%e A339564                                                  (2*3*(4))
%e A339564                                                  ((2)*2*2*3)
%e A339564                                                  (2*2*2*(3))
%t A339564 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A339564 Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]
%Y A339564 The additive version is A000070 (strict: A015723).
%Y A339564 The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).
%Y A339564 Allowing point (1) gives A057567.
%Y A339564 Choosing a position instead of value gives A066637.
%Y A339564 The ordered additive version is A336875.
%Y A339564 A000005 counts divisors.
%Y A339564 A001787 count normal multisets with a selected position.
%Y A339564 A001792 counts compositions with a selected position.
%Y A339564 A006128 counts partitions with a selected position.
%Y A339564 A066186 count strongly normal multisets with a selected position.
%Y A339564 A254577 counts ordered factorizations with a selected position.
%Y A339564 Cf. A007716, A050336, A281113, A281116, A292886, A293627.
%K A339564 nonn
%O A339564 1,4
%A A339564 _Gus Wiseman_, Apr 10 2021