cp's OEIS Frontend

A342086 Number of strict factorizations of divisors of n.

%I A342086 #10 Mar 10 2021 12:23:31
%S A342086 1,2,2,3,2,5,2,5,3,5,2,9,2,5,5,7,2,9,2,9,5,5,2,16,3,5,5,9,2,15,2,10,5,
%T A342086 5,5,18,2,5,5,16,2,15,2,9,9,5,2,25,3,9,5,9,2,16,5,16,5,5,2,31,2,5,9,
%U A342086 14,5,15,2,9,5,15,2,34,2,5,9,9,5,15,2,25,7,5
%N A342086 Number of strict factorizations of divisors of n.
%C A342086 A strict factorization of n is a set of distinct positive integers > 1 with product n.
%H A342086 Robert Israel, <a href="/A342086/b342086.txt">Table of n, a(n) for n = 1..10000</a>
%e A342086 The a(1) = 1 through a(12) = 9 factorizations:
%e A342086   ()  ()   ()   ()   ()   ()     ()   ()     ()   ()     ()    ()
%e A342086       (2)  (3)  (2)  (5)  (2)    (7)  (2)    (3)  (2)    (11)  (2)
%e A342086                 (4)       (3)         (4)    (9)  (5)          (3)
%e A342086                           (6)         (8)         (10)         (4)
%e A342086                           (2*3)       (2*4)       (2*5)        (6)
%e A342086                                                                (12)
%e A342086                                                                (2*3)
%e A342086                                                                (2*6)
%e A342086                                                                (3*4)
%p A342086 sf1:= proc(n,m)
%p A342086   local D,d;
%p A342086   if n = 1 then return 1 fi;
%p A342086   D:= select(`<`,numtheory:-divisors(n) minus {1},m);
%p A342086   add( procname(n/d,d), d= D)
%p A342086 end proc:
%p A342086 sf:= proc(n) option remember; sf1(n,n+1) end proc:f:= proc(n) local d; add(sf(d),d=numtheory:-divisors(n)) end proc:map(f, [$1..100]); # _Robert Israel_, Mar 10 2021
%t A342086 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A342086 Table[Sum[Length[Select[facs[k],UnsameQ@@#&]],{k,Divisors[n]}],{n,30}]
%Y A342086 A version for partitions is A026906 (strict partitions of 1..n).
%Y A342086 A version for partitions is A036469 (strict partitions of 0..n).
%Y A342086 A version for partitions is A047966 (strict partitions of divisors).
%Y A342086 The non-strict version is A057567.
%Y A342086 A000005 counts divisors, with sum A000203.
%Y A342086 A000009 counts strict partitions.
%Y A342086 A001055 counts factorizations, with strict case A045778.
%Y A342086 A001221 counts prime divisors, with sum A001414.
%Y A342086 A001222 counts prime-power divisors.
%Y A342086 A005117 lists squarefree numbers.
%Y A342086 Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.
%K A342086 nonn
%O A342086 1,2
%A A342086 _Gus Wiseman_, Mar 05 2021