OFFSET
1,2
COMMENTS
a(17) <= 214555365177753600. a(18) = 3423420, a(20) = 4084080, a(24) = 6126120, a(30) = 46558512, a(32) = 38798760. - Hiroaki Yamanouchi, Oct 03 2014
EXAMPLE
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.
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);
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Paolo P. Lava, Feb 12 2013
EXTENSIONS
a(1) corrected and a(11), a(13) and a(14) added by Hiroaki Yamanouchi, Oct 03 2014
STATUS
approved