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

1,2

Links

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

Formula

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

Example

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.

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 *)

Prog

(PARI) A138010(n) = sumdiv(n, d, if(!(numdiv(n)%d), 1, 0)); \\ Antti Karttunen, May 25 2017
(Scheme) (define (A138010 n) (A000005 (gcd n (A000005 n)))) ;; 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
(Magma) [#Divisors( Gcd(n, #Divisors(n))):n in [1..120]]; // Marius A. Burtea, Aug 03 2019

Crossrefs

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

Sequence in context: A349658 A345992 A350333 * A206487 A209062 A167204

Adjacent sequences: A138007 A138008 A138009 * A138011 A138012 A138013

Keyword

nonn

Author

Leroy Quet, Feb 27 2008

Extensions

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

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

Status

approved