A138011 -id:A138011 - cp's OEIS frontend

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

Leroy Quet, Feb 27 2008

nonn

			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.

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

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

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

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

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 *)

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

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

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

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

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

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

A138012 a(n) = number of positive divisors, k, of n where d(k) divides n (where d(k) = number of positive divisors of k, A000005).

Original entry on oeis.org

1, 2, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 1, 3, 1, 5, 1, 4, 1, 3, 1, 8, 1, 3, 2, 4, 1, 4, 1, 3, 1, 3, 1, 9, 1, 3, 1, 6, 1, 4, 1, 4, 2, 3, 1, 8, 1, 3, 1, 4, 1, 5, 1, 6, 1, 3, 1, 11, 1, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 12, 1, 3, 2, 4, 1, 4, 1, 8, 2, 3, 1, 11, 1, 3, 1, 6, 1, 7, 1, 4, 1, 3, 1, 10, 1, 3, 2, 4, 1, 4, 1, 6

Offset: 1

Leroy Quet, Feb 27 2008

nonn

First occurrence of k: 1, 2, 6, 20, 18, 12, 90, 24, 36, 96, 60, 72, 5670, 972, 120, 336, 180, 420, 540, 240, 600, 2352, 360, 480, 900, 3000, 840, 1080, 1260, 720, ..., . - Robert G. Wilson v

			10 has 4 divisors (1,2,5,10). The number of divisors of each of these divisors of 10 form the sequence (1,2,2,4). Of these, three divide 10: 1,2,2. So a(10) = 3.

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

Cf. A000005, A138010, A138011.

with(numtheory): a:=proc(n) local div, ct, j: div:=divisors(n): ct:=0: for j to tau(n) do if `mod`(n, tau(div[j]))=0 then ct:=ct+1 else end if end do: ct end proc: seq(a(n),n=1..80); # Emeric Deutsch, Mar 14 2008

Table[Length[Select[Divisors[n], Mod[n, Length[Divisors[ # ]]] == 0 &]], {n,1,100}] (* Stefan Steinerberger *)
f[n_] := Count[Mod[n, DivisorSigma[0, Divisors@n]], 0]; Array[f, 104] (* Robert G. Wilson v *)

A138012(n) = sumdiv(n,d,if(!(n%numdiv(d)), 1, 0)); \\ Antti Karttunen, May 25 2017

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

More terms from Stefan Steinerberger and Robert G. Wilson v, Feb 29 2008

A183008 a(n) = 24p(n) = 24A000041(n).

Original entry on oeis.org

24, 24, 48, 72, 120, 168, 264, 360, 528, 720, 1008, 1344, 1848, 2424, 3240, 4224, 5544, 7128, 9240, 11760, 15048, 19008, 24048, 30120, 37800, 46992, 58464, 72240, 89232, 109560, 134496, 164208, 200376, 243432, 295440, 357192, 431448, 519288, 624360, 748440

Offset: 0

Omar E. Pol, Jan 22 2011

a(n) is also the area of one of the faces of the rectangular cuboid which is a three-dimensional version of the shell model of partitions of n. The areas of the other faces are A008606(n)=24*n and A066186(n)=n*p(n), where p(n) is the number of partitions of n. See A135010 for more information.

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*11 = 264.

Cf. A000041, A008606, A018253, A066186, A135010, A138121, A138011.

24 PartitionsP[Range[0,40]] (* Harvey P. Dale, Sep 17 2024 *)

cp's OEIS Frontend

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))).

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A138012 a(n) = number of positive divisors, k, of n where d(k) divides n (where d(k) = number of positive divisors of k, A000005).

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A183008 a(n) = 24p(n) = 24A000041(n).

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cp's OEIS Frontend

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))).

Views

Author

Keywords

Examples

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Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Scheme

Formula

Extensions

A138012 a(n) = number of positive divisors, k, of n where d(k) divides n (where d(k) = number of positive divisors of k, A000005).

Views

Author

Keywords

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Examples

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Programs

Maple

Mathematica

PARI

Python

Extensions

A183008 a(n) = 24*p(n) = 24*A000041(n).

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Author

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Examples

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Programs

Mathematica

A183008 a(n) = 24p(n) = 24A000041(n).