A331200 - OEIS

%I #12 Jan 17 2020 14:01:01

%S 1,6,12,24,48,60,96,120,180,240,360,480,720,840,1080,1260,1440,1680,

%T 2160,2520,3360,4320,5040,7560,8640,10080,15120,20160,25200,30240,

%U 40320,45360,50400,55440,60480,75600,90720,100800,110880,120960,151200,181440,221760

%N Least number with each record number of factorizations into distinct factors > 1.

%C First differs from A330997 in lacking 64.

%H Giovanni Resta, <a href="/A331200/b331200.txt">Table of n, a(n) for n = 1..122</a>

%H 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 Strict factorizations of the initial terms:

%e   ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)

%e       (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)

%e              (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)

%e                     (2*12)   (3*16)   (4*15)    (4*24)    (4*30)

%e                     (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)

%e                              (2*3*8)  (6*10)    (8*12)    (6*20)

%e                              (2*4*6)  (2*5*6)   (2*6*8)   (8*15)

%e                                       (3*4*5)   (3*4*8)   (10*12)

%e                                       (2*3*10)  (2*3*16)  (3*5*8)

%e                                                 (2*4*12)  (4*5*6)

%e                                                           (2*3*20)

%e                                                           (2*4*15)

%e                                                           (2*5*12)

%e                                                           (2*6*10)

%e                                                           (3*4*10)

%e                                                           (2*3*4*5)

%t nn=1000;

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

%t qv=Table[Length[strfacs[n]],{n,nn}];

%t Table[Position[qv,i][[1,1]],{i,Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]}]

%Y A subset of A330997.

%Y All terms belong to A025487.

%Y This is the strict version of highly factorable numbers A033833.

%Y The corresponding records are A331232(n) = A045778(a(n)).

%Y Factorizations are A001055 with image A045782 and complement A330976.

%Y Strict factorizations are A045778 with image A045779 and complement A330975.

%Y The least number with n strict factorizations is A330974(n).

%Y The least number with A045779(n) strict factorizations is A045780(n)

%Y Cf. A045783, A325238, A330972, A330973, A331023/A331024, A331201.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jan 12 2020

%E a(37) and beyond from _Giovanni Resta_, Jan 17 2020