cp's OEIS Frontend

A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

%I A331198 #13 Apr 27 2020 09:19:18
%S A331198 128,2187,10368,34992,78125,80000,307328,823543,1250000,1366875,
%T A331198 1874048,3655808,5250987,6328125,10690688,13176688,16681088,19487171,
%U A331198 32019867,35819648,62462907,62748517,66706983,90531968,118210688,182660427,187578125,239892608,285012027
%N A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).
%C A331198 Contains p^7 for all primes p.
%e A331198 The 15 factorizations and 5 strict factorizations of 2187:
%e A331198   (2187)           (2187)
%e A331198   (27*81)          (27*81)
%e A331198   (3*729)          (3*729)
%e A331198   (9*243)          (9*243)
%e A331198   (3*9*81)         (3*9*81)
%e A331198   (9*9*27)
%e A331198   (3*27*27)
%e A331198   (3*3*243)
%e A331198   (3*9*9*9)
%e A331198   (3*3*3*81)
%e A331198   (3*3*9*27)
%e A331198   (3*3*3*9*9)
%e A331198   (3*3*3*3*27)
%e A331198   (3*3*3*3*3*9)
%e A331198   (3*3*3*3*3*3*3)
%t A331198 facsm[n_]:=facsm[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsm[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A331198 Select[Range[100000],3==Length[facsm[#]]/Length[Select[facsm[#],UnsameQ@@#&]]&]
%Y A331198 Factorizations are A001055.
%Y A331198 Strict factorizations are A045778.
%Y A331198 Taking "twice" instead of "three times" gives A001248.
%Y A331198 Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.
%K A331198 nonn
%O A331198 1,1
%A A331198 _Gus Wiseman_, Jan 12 2020
%E A331198 a(7)-(10) from _Alois P. Heinz_, Jan 17 2020
%E A331198 a(11)-a(29) from _Giovanni Resta_, Jan 20 2020