A222084 Number of the least divisors of n whose LCM is equal to n.

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 4, 2, 3, 3, 5, 2, 5, 2, 4, 3, 3, 2, 6, 3, 3, 4, 4, 2, 4, 2, 6, 3, 3, 3, 6, 2, 3, 3, 5, 2, 5, 2, 4, 4, 3, 2, 8, 3, 5, 3, 4, 2, 7, 3, 5, 3, 3, 2, 5, 2, 3, 4, 7, 3, 5, 2, 4, 3, 4, 2, 7, 2, 3, 5, 4, 3, 5, 2, 7, 5, 3, 2, 6, 3, 3, 3

Offset

1,2

Comments

If we write n as the product of its prime factors, n = p1^a1*p2^a2*p3^a3*...*pr^ar, then tau#(n) gives the number of divisors from 1 to max(p1^a1, p2^a2, p3^a3, ..., pr^ar).

In general tau#(n) <= tau(n).

Also, tau#(n) = tau(n) is A000961, tau#(n) < tau(n) is A024619.

For any prime number p tau(p) = tau#(p) = 2.

tau#(n) = 3 only for semiprimes (A001358).

Links

Paolo P. Lava, Table of n, a(n) for n = 1..1000

Example

For n=40, the divisors are (1, 2, 4, 5, 8, 10, 20, 40), so tau(40)=8.
lcm(1, 2, 4, 5, 8) = 40, but lcm(1, 2, 4, 5) = 20 < 40, so tau#(40)=5.

Maple

with(numtheory);
A222084:=proc(q)
local a, b, c, j, n; print(1);
for n from 2 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; print(j); od; end:
A222084(100000);

Mathematica

Table[Count[ Divisors[n] , q_Integer /; q <= Max[Power @@@ FactorInteger[n]]], {n, 87}] (* Wouter Meeussen, Feb 09 2013 *)

Prog

(PARI) a(n) = {my(d = divisors(n), k = 1); while (lcm(vector(k, j, d[j])) != n, k++); k; } \\ Michel Marcus, Mar 13 2018

Crossrefs

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

Sequence in context: A353861 A073093 A326196 * A327394 A088873 A085082

Adjacent sequences: A222081 A222082 A222083 * A222085 A222086 A222087

Keyword

nonn

Author

Paolo P. Lava, Feb 07 2013

Status

approved