A222086 a(n) is the least number k for which A000005(k)/A222084(k) = n.

1, 30, 360, 840, 11088, 18018, 1713600, 32760, 327600, 350064, 39437798400, 180180, 8532905472000, 47361600, 720720, 1750320

Offset: 1

Author

Paolo P. Lava, Feb 12 2013

Keywords

• nonn
• more

Comments

a(17) <= 214555365177753600. a(18) = 3423420, a(20) = 4084080, a(24) = 6126120, a(30) = 46558512, a(32) = 38798760. - Hiroaki Yamanouchi, Oct 03 2014

Examples

			For k=18018, tau(k)=48: the 48 divisors of k are 1, 2, 3, 6, 7, 9, 11, 13, 14, 18, 21, 22, 26, 33, 39, 42, 63, 66, 77, 78, 91, 99, 117, 126, 143, 154, 182, 198, 231, 234, 273, 286, 429, 462, 546, 693, 819, 858, 1001, 1287, 1386, 1638, 2002, 2574, 3003, 6006, 9009, 18018.
The least common multiple of the first 8 divisors, (1, 2, 3, 6, 7, 9, 11, 13), is again 18018, but the least common multiple of the first 7 divisors, (1, 2, 3, 6, 7, 9, 11), is less than 18018.
Since tau#(k)=8 (see A222084 for the definition of tau#(n)), tau(k)/tau#(k) = 48/8 = 6, and since 18018 is the minimum number k to have this ratio, a(6)=18018.
		

Crossrefs

Cf. A000005, A000961, A001358, A003418, A005179, A024619, A034444, A077610, A222084, A222085.

Programs

Maple

with(numtheory);
A222086:=proc(q)
local a,b,c,d,j,n,t,v;
v:=array(1..100); for j from 1 to 100 do v[j]:=0; od; t:=0;
for n from 1 to q do
  a:=ifactors(n)[2]; b:=nops(a); c:=0;
  for j from 1 to b do if a[j][1]^a[j][2]>c then c:=a[j][1]^a[j][2]; fi; od;
  a:=op(sort([op(divisors(n))])); b:=nops(divisors(n));
  for j from 1 to b do if a[j]=c then break; fi; od;
  if type(tau(n)/j,integer)  then if tau(n)/j=t+1
       then t:=t+1; lprint(t,n); while v[t+1]>0 do t:=t+1; lprint(t,v[t]); od;
       else if tau(n)/j>t+1 then if v[tau(n)/j]=0 then v[tau(n)/j]:=n; fi; fi;
fi; fi; od; end:
A222086(1000000000000000);

Extensions

a(1) corrected and a(11), a(13) and a(14) added by Hiroaki Yamanouchi, Oct 03 2014