cp's OEIS Frontend

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

%I A336424 #13 Sep 02 2020 23:04:37
%S A336424 1,1,1,2,1,1,1,3,2,1,1,3,1,1,1,5,1,3,1,3,1,1,1,5,2,1,3,3,1,1,1,7,1,1,
%T A336424 1,6,1,1,1,5,1,1,1,3,3,1,1,9,2,3,1,3,1,5,1,5,1,1,1,4,1,1,3,11,1,1,1,3,
%U A336424 1,1,1,11,1,1,3,3,1,1,1,9,5,1,1,4,1,1
%N A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).
%C A336424 A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.
%e A336424 The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
%e A336424   2  4    8      5*12     16       4*9      32         48
%e A336424      2*2  2*4    3*20     4*4      3*12     4*8        4*12
%e A336424           2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
%e A336424                  2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
%e A336424                           2*2*2*2  2*2*9    2*2*8      2*24
%e A336424                                    2*2*3*3  2*2*2*4    2*3*8
%e A336424                                             2*2*2*2*2  2*2*12
%e A336424                                                        2*2*3*4
%e A336424                                                        2*2*2*2*3
%t A336424 facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
%t A336424 Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]
%Y A336424 A327523 is the case when n is restricted to belong to A130091 also.
%Y A336424 A001055 counts factorizations.
%Y A336424 A007425 counts divisors of divisors.
%Y A336424 A045778 counts strict factorizations.
%Y A336424 A074206 counts ordered factorizations.
%Y A336424 A130091 lists numbers with distinct prime multiplicities.
%Y A336424 A181796 counts divisors with distinct prime multiplicities.
%Y A336424 A253249 counts nonempty chains of divisors.
%Y A336424 A281116 counts factorizations with no common divisor.
%Y A336424 A302696 lists numbers whose prime indices are pairwise coprime.
%Y A336424 A305149 counts stable factorizations.
%Y A336424 A320439 counts factorizations using A289509.
%Y A336424 A327498 gives the maximum divisor with distinct prime multiplicities.
%Y A336424 A336500 counts divisors of n in A130091 with quotient also in A130091.
%Y A336424 A336568 = not a product of two numbers with distinct prime multiplicities.
%Y A336424 A336569 counts maximal chains of elements of A130091.
%Y A336424 A337256 counts chains of divisors.
%Y A336424 Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.
%K A336424 nonn
%O A336424 1,4
%A A336424 _Gus Wiseman_, Aug 03 2020