A138010 a(n) is the number of positive divisors of n that divide d(n), where d(n) is the number of positive divisors of n, A000005(n); a(n) also equals d(gcd(n, d(n))).

1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 2, 1, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4

Offset: 1

Author

Leroy Quet, Feb 27 2008

Keywords

• nonn

Examples

			12 has 6 divisors (1,2,3,4,6,12). Those divisors of 12 that divide 6 are 1,2,3,6. Since there are 4 of these, then a(12) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A009191, A124315, A138011, A138012.

Programs

Magma

[#Divisors( Gcd(n,#Divisors(n))):n in [1..120]]; // Marius A. Burtea, Aug 03 2019

Maple

with(numtheory): a:=proc(n) local div,c,j: div:=divisors(n): c:=0: for j to tau(n) do if `mod`(tau(n), div[j])=0 then c:=c+1 else end if end do: c end proc: seq(a(n),n=1..90); # Emeric Deutsch, Mar 02 2008

Mathematica

Table[Length[Select[Divisors[n], Mod[Length[Divisors[n]], # ] == 0 &]], {n,1,100}] (* Stefan Steinerberger, Feb 29 2008 *)
Table[Count[DivisorSigma[0,n]/Divisors[n],?IntegerQ],{n,120}] (* _Harvey P. Dale, May 31 2019 *)

PARI

A138010(n) = sumdiv(n,d,if(!(numdiv(n)%d), 1, 0)); \\ Antti Karttunen, May 25 2017

Python

from sympy import divisors, divisor_count
def a(n): return sum([ 1*(divisor_count(n)%d==0) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

Scheme

(define (A138010 n) (A000005 (gcd n (A000005 n)))) ;; Antti Karttunen, May 25 2017

Formula

a(n) = A000005(A009191(n)). [From the alternative description.] - Antti Karttunen, May 25 2017

Extensions

More terms from Stefan Steinerberger and Emeric Deutsch, Feb 29 2008

Further extended (to 120 terms) by Antti Karttunen, May 25 2017