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A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

%I A050322 #19 Jan 13 2020 13:54:31
%S A050322 1,1,2,2,3,4,5,7,5,7,9,12,11,11,16,19,21,15,29,26,30,15,31,38,22,47,
%T A050322 52,45,36,57,64,30,77,98,67,74,97,66,105,42,109,118,92,109,171,97,141,
%U A050322 162,137,165,56,212,181,52,198,189,289,139,250,257,269,254,77,382,267
%N A050322 Number of factorizations indexed by prime signatures: A001055(A025487).
%C A050322 For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).
%C A050322 Is a(k) = A000110(k) for A025487(m) = A002110(k)?
%H A050322 R. J. Mathar and Michael De Vlieger, <a href="/A050322/b050322.txt">Table of n, a(n) for n = 1..5000</a> (First 300 terms from _R. J. Mathar_)
%H A050322 R. E. Canfield, P. Erdős and C. Pomerance, <a href="http://math.dartmouth.edu/~carlp/PDF/paper39.pdf">On a Problem of Oppenheim concerning "Factorisatio Numerorum"</a>, J. Number Theory 17 (1983), 1-28.
%H A050322 Jun Kyo Kim, <a href="https://doi.org/10.1006/jnth.1998.2238">On highly factorable numbers</a>, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
%e A050322 From _Gus Wiseman_, Jan 13 2020: (Start)
%e A050322 The a(1) = 1 through a(11) = 9 factorizations:
%e A050322   {}  2  4    6    8      12     16       24       30     32         36
%e A050322          2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
%e A050322                    2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
%e A050322                           2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
%e A050322                                  2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
%e A050322                                           2*3*4           2*2*2*4    2*2*9
%e A050322                                           2*2*2*3         2*2*2*2*2  2*3*6
%e A050322                                                                      3*3*4
%e A050322                                                                      2*2*3*3
%e A050322 (End)
%p A050322 A050322 := proc(n)
%p A050322     A001055(A025487(n)) ;
%p A050322 end proc: # _R. J. Mathar_, May 25 2017
%t A050322 c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* _Michael De Vlieger_, Jul 10 2017, after _Dean Hickerson_ at A001055 *)
%t A050322 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A050322 Length/@facs/@First/@GatherBy[Range[1000],If[#==1,{},Sort[Last/@FactorInteger[#]]]&] (* _Gus Wiseman_, Jan 13 2020 *)
%Y A050322 Cf. A000041, A000079, A000110, A001055, A002110, A025487.
%Y A050322 The version indexed by unsorted prime signature is A331049.
%Y A050322 The version indexed by prime shadow (A181819, A181821) is A318284.
%Y A050322 This sequence has range A045782 (same as A001055).
%Y A050322 Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.
%K A050322 nonn
%O A050322 1,3
%A A050322 _Christian G. Bower_, Oct 15 1999