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Number of strict factorizations of divisors of n.
15

%I #10 Mar 10 2021 12:23:31

%S 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 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 14,5,15,2,9,5,15,2,34,2,5,9,9,5,15,2,25,7,5

%N Number of strict factorizations of divisors of n.

%C A strict factorization of n is a set of distinct positive integers > 1 with product n.

%H Robert Israel, <a href="/A342086/b342086.txt">Table of n, a(n) for n = 1..10000</a>

%e The a(1) = 1 through a(12) = 9 factorizations:

%e () () () () () () () () () () () ()

%e (2) (3) (2) (5) (2) (7) (2) (3) (2) (11) (2)

%e (4) (3) (4) (9) (5) (3)

%e (6) (8) (10) (4)

%e (2*3) (2*4) (2*5) (6)

%e (12)

%e (2*3)

%e (2*6)

%e (3*4)

%p sf1:= proc(n,m)

%p local D,d;

%p if n = 1 then return 1 fi;

%p D:= select(`<`,numtheory:-divisors(n) minus {1},m);

%p add( procname(n/d,d), d= D)

%p end proc:

%p 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 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Sum[Length[Select[facs[k],UnsameQ@@#&]],{k,Divisors[n]}],{n,30}]

%Y A version for partitions is A026906 (strict partitions of 1..n).

%Y A version for partitions is A036469 (strict partitions of 0..n).

%Y A version for partitions is A047966 (strict partitions of divisors).

%Y The non-strict version is A057567.

%Y A000005 counts divisors, with sum A000203.

%Y A000009 counts strict partitions.

%Y A001055 counts factorizations, with strict case A045778.

%Y A001221 counts prime divisors, with sum A001414.

%Y A001222 counts prime-power divisors.

%Y A005117 lists squarefree numbers.

%Y Cf. A001227, A050320, A340101, A340596, A340654, A340655, A340853, A341596, A341673, A341674, A342097.

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 05 2021