A009191 -id:A009191 - cp's OEIS frontend

A322979 a(n) = Sum A009191(d) over the divisors d of n, where A009191(x) = gcd(x, A000005(x)), and A000005(x) gives the number of divisors of x.

1, 3, 2, 4, 2, 6, 2, 8, 5, 6, 2, 13, 2, 6, 4, 9, 2, 15, 2, 9, 4, 6, 2, 25, 3, 6, 6, 9, 2, 12, 2, 11, 4, 6, 4, 31, 2, 6, 4, 21, 2, 12, 2, 9, 10, 6, 2, 28, 3, 9, 4, 9, 2, 18, 4, 21, 4, 6, 2, 33, 2, 6, 10, 12, 4, 12, 2, 9, 4, 12, 2, 55, 2, 6, 8, 9, 4, 12, 2, 32, 7, 6, 2, 33, 4, 6, 4, 21, 2, 30, 4, 9, 4, 6, 4, 42, 2, 9, 10, 13, 2, 12, 2, 21, 8

Offset: 1

Antti Karttunen, Jan 05 2019

nonn

Inverse Möbius transform of A009191.

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A009191.

A009191(n) = gcd(n, numdiv(n));
A322979(n) = sumdiv(n,d,A009191(d));

a(n) = Sum_{d|n} A009191(d).

A286540 a(n) = gcd(A259934(n), A259934(n-1)) = A009191(A259934(n)).

Original entry on oeis.org

2, 2, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 1, 1, 1, 3, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 6, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 8, 8, 24, 18, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 6

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..64800 (computed from the b-file of A259934 provided by Robert Israel)

Cf. A009191, A259934, A259935.

(define (A286540 n) (gcd (A259934 n) (A259934 (- n 1))))
(define (A286540 n) (gcd (A259934 n) (A259935 n)))

a(n) = gcd(A259934(n), A259934(n-1)) = gcd(A259934(n), A259935(n)).

a(n) = A009191(A259934(n)).

A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

1, 3, 1, 1, 1, 23, 1, 10, 6, 5, 1, 42, 1, 5, 4, 1, 1, 34, 1, 5, 1, 5, 1, 179, 1, 5, 1, 408, 1, 23, 1, 3, 4, 5, 1, 45, 1, 5, 1, 144, 1, 23, 1, 12, 13, 5, 1, 12, 1, 3, 4, 5, 1, 23, 1, 113, 1, 5, 1, 265, 1, 5, 6, 1, 1, 23, 1, 5, 4, 5, 1, 103, 1, 5, 6, 12, 1, 23, 1, 65, 1, 5, 1, 753, 1, 5, 4, 63, 1, 259, 22, 12, 1, 5, 11, 265, 1, 3, 13, 1, 1, 23, 1, 44, 4, 5, 1

Offset: 1

Antti Karttunen, May 21 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A009191, A009194, A286594.

A009191(n) = gcd(n, numdiv(n));
A009194(n) = gcd(n, sigma(n));
A286591(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n));

from sympy import divisor_sigma, divisor_count, gcd
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def a(n): return T(gcd(n, divisor_count(n)), gcd(n, divisor_sigma(n))) # Indranil Ghosh, May 22 2017

(define (A286591 n) (* (/ 1 2) (+ (expt (+ (A009191 n) (A009194 n)) 2) (- (A009191 n)) (- (* 3 (A009194 n))) 2)))

a(n) = (1/2)*(2 + ((A009191(n)+A009194(n))^2) - A009191(n) - 3*A009194(n)).

A322974 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0, f(2) = 1, f(n) = 3 if A009191(n) == 1 and f(n) = A049820(n) for all other numbers > 2.

Original entry on oeis.org

1, 2, 3, 3, 3, 4, 3, 5, 6, 6, 3, 6, 3, 7, 3, 3, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 12, 3, 12, 3, 13, 3, 14, 3, 15, 3, 16, 3, 17, 3, 16, 3, 18, 19, 20, 3, 18, 3, 21, 3, 22, 3, 22, 3, 23, 3, 24, 3, 23, 3, 25, 26, 3, 3, 25, 3, 27, 3, 27, 3, 28, 3, 29, 30, 29, 3, 29, 3, 29, 3, 31, 3, 32, 3, 33, 3, 34, 3, 31, 3, 35, 3, 36, 3, 37, 3, 38, 39, 3, 3, 40, 3, 41, 3

Offset: 1

Antti Karttunen, Jan 05 2019

nonn

For all i, j:
  A322810(i) = A322810(j) => a(i) = a(j),
  a(i) = a(j) => A323073(i) = A323073(j).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A009191, A049820, A323073.

Cf. also A319337, A322810, A322980.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A322974aux(n) = if(n<=2,n-1,my(u=(n-numdiv(n))); if(1==gcd(n,u),3,u));
v322974 = rgs_transform(vector(up_to,n,A322974aux(n)));
A322974(n) = v322974[n];

A319337 Filter sequence combining gcd(n,tau(n)) (= A009191) with the prime signature of n (A046523).

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 3, 6, 7, 5, 3, 8, 3, 5, 9, 10, 3, 8, 3, 11, 9, 5, 3, 12, 4, 5, 13, 11, 3, 14, 3, 15, 9, 5, 9, 16, 3, 5, 9, 12, 3, 14, 3, 11, 17, 5, 3, 18, 4, 11, 9, 11, 3, 19, 9, 12, 9, 5, 3, 20, 3, 5, 17, 21, 9, 14, 3, 11, 9, 14, 3, 22, 3, 5, 17, 11, 9, 14, 3, 23, 10, 5, 3, 20, 9, 5, 9, 12, 3, 24, 9, 11, 9, 5, 9, 25, 3, 11, 17, 26, 3, 14, 3, 12, 27

Offset: 1

Antti Karttunen, Sep 24 2018

nonn

Restricted growth sequence transform of ordered pair [A009191(n), A046523(n)].

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009191, A046523, A101296, A300230, A300240, A305801, A319338.

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A009191(n) = gcd(n, numdiv(n));
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p=0); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };
v319337 = rgs_transform(vector(up_to,n,[A009191(n),A046523(n)]));
A319337(n) = v319337[n];

A327715 a(0) = 0; for n >= 1, a(n) = 1 + a(n-A009191(n)).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 8, 9, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 12, 13, 13, 14, 14, 15, 15, 16, 16, 16, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 19, 20, 20, 21, 19, 20, 20, 20, 21, 22, 22, 23, 23, 24, 24, 25, 20, 21, 21, 21, 22, 23, 23, 24, 25, 26

Offset: 0

Ctibor O. Zizka, Sep 23 2019

nonn

Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k with k - gcd(k,d(k)), where d(k) is the number of divisors of k (A000005).

Empirically: n/log(n) <= a(n) <= n/log(n) + 2*log(n).

			a(6) = 1 + a(6-gcd(6,4)) = 1 + a(4) = 2 + a(4-gcd(4,3)) = 2 + a(3) = 3 + a(3-gcd(3,2)) = 3 + a(2) = 4 + a(2-gcd(2,2)) = 4 + a(0) = 4.

Cf. A000005, A009191, A155043.

a(n) = if (n==0, 0, 1 + a(n - gcd(n, numdiv(n)))); \\ Michel Marcus, Sep 25 2019

A009194 a(n) = gcd(n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2

Offset: 1

David W. Wilson

nonn

LCM of common divisors of n and sigma(n). It equals n if n is multiply perfect (A007691). - Labos Elemer, Aug 14 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. Volume 60, Issue 1 (2011), 199-214.

Cf. A000203, A003624, A007691, A014567, A017666, A063906, A069059, A073802, A082062, A179931, A205523, A216793 (positions of records), A234367, A249917.

Cf. also A009191, A009205, A009242, A274382, A286591, A286594.

a009194 n = gcd (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

Table[GCD[n,DivisorSigma[1,n]],{n,110}] (* Harvey P. Dale, Aug 23 2015 *)

a(n) = gcd(n, sigma(n)); \\ Michel Marcus, Oct 23 2013

A000005(a(n)) = A073802(n). - Reinhard Zumkeller, Mar 12 2010
A006530(a(n)) = A082062(n). - Reinhard Zumkeller, Jul 10 2011
a(A014567(n)) = 1; A069059(a(n)) > 1. - Reinhard Zumkeller, Mar 23 2013
a(n) = n/A017666(n). - Antti Karttunen, May 22 2017

A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 11, 13, 15, 16, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 47, 49, 51, 53, 55, 57, 59, 61, 64, 65, 67, 69, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 100, 101, 103, 105, 107, 109, 111, 113, 115, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Labos Elemer

Numbers k such that tau(k)^phi(k) == 1 (mod k), where tau(k) is the number of divisors of k (A000005) and phi(k) is the Euler phi function (A000010). - Michel Lagneau, Nov 20 2012
Density is at least 4/Pi^2 = 0.405... since A056911 is a subsequence, and at most 1/2 since all even numbers in this sequence are squares. The true value seems to be around 0.4504. - Charles R Greathouse IV, Mar 27 2013
They are called anti-tau numbers by Zelinsky (see link) and their density is at least 3/Pi^2 (theorem 57 page 15). - Michel Marcus, May 31 2015
From Amiram Eldar, Feb 21 2021: (Start)
Spiro (1981) proved that the number of terms of this sequence that do not exceed x is c * x + O(sqrt(x)*log(x)^3), where 0 < c < 1 is the asymptotic density of this sequence.
The odd numbers whose number of divisors is a power of 2 (the odd terms of A036537) are terms of this sequence. Their asymptotic density is A327839/A076214 = 0.4212451116... which is a better lower bound than 4/Pi^2 for the asymptotic density of this sequence.
A better upper limit than 0.5 can be obtained by considering the subsequence of odd numbers whose 3-adic valuation is not of the form 3*k-1 (i.e., odd numbers without those k with gcd(k, tau(k)) = 3), whose asymptotic density is 6/13 = 0.46153...
The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 5, 49, 459, 4535, 45145, 450710, 4504999, 45043234, 450411577, 4504050401, ... (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Mart Abel, Helena Lauer, and Ellen Redi, About the number of τ-numbers relative to polynomials with integer coefficients, Acta Comment. Univ. Tartu. Math. 25, No. 1, 107-117, 2021.
Claudia A. Spiro, The Frequency with Which an Integral-Valued, Prime-Independent, Multiplicative or Additive Function of n Divides a Polynomial Function of n, Ph. D. Thesis, University of Illinois, Urbana-Champaign, 1981.
Joshua Zelinsky, Tau Numbers: A Partial Proof of a Conjecture and Other Results, Journal of Integer Sequences, Vol. 5 (2002), Article 02.2.8.

Cf. A000005, A000010, A009191, A009230, A036537, A056911, A076214, A327839.

a046642 n = a046642_list !! (n-1)
a046642_list = map (+ 1) $ elemIndices 1 a009191_list
-- Reinhard Zumkeller, Aug 14 2011

Select[ Range[200], CoprimeQ[#, DivisorSigma[0, #]] &] (* Jean-François Alcover, Oct 20 2011 *)

is(n)=gcd(numdiv(n),n)==1 \\ Charles R Greathouse IV, Mar 27 2013

A009191(a(n)) = 1.

A009205 a(n) = gcd(d(n), sigma(n)).

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

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

Cf. A000005, A000203, A009191, A009194, A009278, A064840, A087801, A286360.

Table[GCD[DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Dec 05 2017 *)

A009205(n) = gcd(numdiv(n),sigma(n)); \\ Antti Karttunen, May 22 2017

from math import prod, gcd
from sympy import factorint
def A009205(n):
    f = factorint(n).items()
    return gcd(prod(e+1 for p, e in f),prod((p**(e+1)-1)//(p-1) for p,e in f)) # Chai Wah Wu, Jul 27 2023

a(n) = A064840(n)/A009278(n). - Amiram Eldar, Jan 31 2025

Data section extended to 120 terms by Antti Karttunen, May 22 2017

A090395 Denominator of d(n)/n, where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 3, 4, 5, 3, 7, 2, 3, 5, 11, 2, 13, 7, 15, 16, 17, 3, 19, 10, 21, 11, 23, 3, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 4, 37, 19, 39, 5, 41, 21, 43, 22, 15, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 7, 57, 29, 59, 5, 61, 31, 21, 64, 65, 33, 67, 34, 69, 35, 71, 6, 73, 37, 25, 38

Offset: 1

Ivan_E_Mayle(AT)a_provider.com, Jan 31 2004

The first occurrence of k (if it exists) is studied in A091895.

Sequence A353011 gives indices of "late birds": n such that a(k) > a(n) for all k > n. - M. F. Hasler, Apr 15 2022

			a(6) = 3 because the number of divisors of 6 is 4 and 4 divided by 6 equals 2/3, which has 3 as its denominator.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A090387 (numerators), A091896 (numbers not in this sequence), A353011 (indices of terms such that all subsequent terms are larger).

with(numtheory): seq(denom(tau(n)/n), n=1..75) ; # Zerinvary Lajos, Jun 04 2008

Table[ Denominator[ DivisorSigma[0, n]/n], {n, 1, 80}] (* Robert G. Wilson v, Feb 04 2004 *)

A090395(n) = denominator(numdiv(n)/n); \\ Antti Karttunen, Sep 25 2018

from math import gcd
from sympy import divisor_count
def A090395(n): return n//gcd(n,divisor_count(n)) # Chai Wah Wu, Jun 20 2022

a(n) = n/g with g = A009191(n) = gcd(A000005(n), n). This explains the "rays" in the graph, e.g., g = 1 for odd squarefree n, g = 2 for even semiprimes n = 2p > 4 and n = 4p, p > 3. - M. F. Hasler, Apr 15 2022

More terms from Robert G. Wilson v, Feb 04 2004

cp's OEIS Frontend

A322979 a(n) = Sum A009191(d) over the divisors d of n, where A009191(x) = gcd(x, A000005(x)), and A000005(x) gives the number of divisors of x.

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A286540 a(n) = gcd(A259934(n), A259934(n-1)) = A009191(A259934(n)).

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A286591 Compound filter: a(n) = P(A009191(n), A009194(n)), where P(n,k) is sequence A000027 used as a pairing function.

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A322974 Lexicographically earliest such sequence a that a(i) = a(j) => f(i) = f(j) for all i, j, where f(1) = 0, f(2) = 1, f(n) = 3 if A009191(n) == 1 and f(n) = A049820(n) for all other numbers > 2.

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A319337 Filter sequence combining gcd(n,tau(n)) (= A009191) with the prime signature of n (A046523).

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A327715 a(0) = 0; for n >= 1, a(n) = 1 + a(n-A009191(n)).

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A009194 a(n) = gcd(n, sigma(n)).

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A046642 Numbers k such that k and number of divisors d(k) are relatively prime.

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