A138010 -id:A138010 - cp's OEIS frontend

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 2, 1, 10, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 8, 1

Offset: 1

David W. Wilson

nonn

a(A046642(n)) = 1.
First occurrence of k: 1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, .... Conjecture: each k is present. - Robert G. Wilson v, Mar 27 2013
Conjecture is true. See David A. Corneth's comment in A324553. - Antti Karttunen, Mar 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A000005, A009194, A009195, A009205, A009213, A009230, A049820, A125168, A138010, A286540, A303781, A318459, A319337, A322979, A322980, A323073.

Cf. A046642 (positions of ones), A324553 (position of the first occurrence of each n).

a009191 n = gcd n $ a000005 n
-- Reinhard Zumkeller, May 09 2013, Aug 14 2011

f[n_] := GCD[n, DivisorSigma[0, n]]; Array[f, 105] (* Robert G. Wilson v, Mar 27 2013 *)

a(n)=gcd(numdiv(n),n) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = gcd(n, A000005(n)) = gcd(n, A049820(n)). - Antti Karttunen, Sep 25 2018

A138011 a(n) = number of positive divisors, k, of n where d(k) divides d(n). (d(m) = number of positive divisors of m, A000005).

Original entry on oeis.org

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

Offset: 1

Leroy Quet, Feb 27 2008

nonn

			12 has 6 divisors (1,2,3,4,6,12). The number of divisors of each of these divisors of 12 form the sequence (1,2,2,3,4,6). Of these, five divide d(12)=6: 1,2,2,3,6. So a(12) = 5.

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

Cf. A000005, A138010, A138012.

Table[Length[Select[Divisors[n], Mod[Length[Divisors[n]], Length[Divisors[ # ]]] == 0 &]], {n, 1, 100}] (* Stefan Steinerberger, Feb 29 2008 *)

A138011(n) = sumdiv(n,d,if(!(numdiv(n)%numdiv(d)), 1, 0)); \\ Antti Karttunen, May 25 2017

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

More terms from Stefan Steinerberger, Feb 29 2008

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

cp's OEIS Frontend

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

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A138011 a(n) = number of positive divisors, k, of n where d(k) divides d(n). (d(m) = number of positive divisors of m, A000005).

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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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Examples

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Programs

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PARI

Python

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