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A222084 Number of the least divisors of n whose LCM is equal to n. 6
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 (list; graph; refs; listen; history; text; internal format)
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: A299990 A175193 A073093 * A088873 A085082 A181796

Adjacent sequences:  A222081 A222082 A222083 * A222085 A222086 A222087

KEYWORD

nonn

AUTHOR

Paolo P. Lava, Feb 07 2013

STATUS

approved

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Last modified May 23 11:00 EDT 2018. Contains 304466 sequences. (Running on oeis4.)