A009230 -id:A009230 - cp's OEIS frontend

A033950 Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, 328, 344, 348, 360, 372, 376, 384, 396, 424, 441, 444, 448, 450, 468, 472, 480, 488, 492, 504, 516, 536

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk)

Kennedy and Cooper show that this sequence has density zero.
Spiro showed more precisely that the number of refactorable numbers less than x is asymptotic to (x/sqrt(log x))(log(log x))^(-1+o(1)). - David Eppstein, Aug 25 2014
Numbers k such that the equation gcd(k,x) = tau(k) has solutions. - Benoit Cloitre, Jun 10 2002
Refactorable numbers are the fixed points of A009230. - Labos Elemer, Nov 18 2002
Let ref(n) denote the characteristic function of the refactorable numbers. Then ref(n) = 1 + floor(n/d(n)) - ceiling(n/d(n)), where d(n) is the number of divisors of n. - Wesley Ivan Hurt, Jan 09 2013, Feb 15 2013
An odd number with an even number of divisors cannot be in the sequence by definition. Therefore all odd terms are squares (A000290). - Ivan N. Ianakiev, Aug 25 2013
A054008(k) = k mod A000005(k). - Reinhard Zumkeller, Sep 17 2014
The only squarefree terms are 1 and 2: if x is a squarefree number that is a product of n distinct primes, its number of divisors is 2^n, so x is refactorable if it contains 2^n as a factor, but that makes it nonsquarefree unless n = 0, 1, hence x = 1, 2. - Waldemar Puszkarz, Jun 10 2016
Every positive integer k occurs as tau(m) for some m in the sequence. If the factorization of k is Product p_i^e_i, then Product p_i^(p_i^e_i-1) has the specified property. For k prime, this is the only such number. - Franklin T. Adams-Watters, Jan 14 2017
Zelinsky (2002) proved that for any j > 0 and for sufficiently large m the number of terms not exceeding m is > j*pi(m), where pi(m) = A000720(m). - Amiram Eldar, Feb 20 2021
Numbers m such that the ratio (number of non-divisors of m)/(number of divisors of m) = A049820(m)/A000005(m) is an integer. - Michel Lagneau, Apr 04 2025

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B12, pp. 102-103.
New Scientist, Sep 05 1998, p. 17, para. 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Kushagr Ahuja, Patrick Lei and Dylan Pentland, Tau ideals in number fields, PROMYS 2017.
Alan Bundy, Simon Colton and Toby Walsh, HR - A system for Machine Discovery in Finite Algebras, ECAI 1998.
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics.
Robert E. Kennedy and Curtis N. Cooper, Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, Vol. 13, No. 2 (1990), pp. 383-386.
Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory, Vol. 21, No. 1 (1985), pp. 81-100.
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, A000720, A039819, A036762, A051278, A051279, A051280, A036763.
Cf. A235353 (subsequence).
Cf. A054008, A281188.

a033950 n = a033950_list !! (n-1)
a033950_list = [x | x <- [1..], x `mod` a000005 x == 0]
-- Reinhard Zumkeller, Dec 28 2011

[ n: n in [1..540] | n mod #Divisors(n) eq 0 ]; // Klaus Brockhaus, Apr 29 2009

with(numtheory):
A033950 := proc(n)
    option remember:
    local k:
    if n=1 then
        return 1:
    else
        for k from procname(n-1)+1 do
            if type(k/tau(k), integer) then
                return k:
            end if:
        end do:
    end if:
end proc:
seq(A033950(n), n=1..56); # Nathaniel Johnston, May 04 2011

Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 1000}]
Select[ Range[559], Mod[ #, DivisorSigma[0, # ]] == 0 &]
Select[Range[550], Divisible[ #, DivisorSigma[0, # ]]&] (* Waldemar Puszkarz, Jun 10 2016 *)

isA033950(n)=n%numdiv(n)==0 \\ Charles R Greathouse IV, Jun 10 2011

from sympy import divisor_count
print([n for n in range(1, 1001) if not n % divisor_count(n)]) # Indranil Ghosh, May 03 2017

More terms from Erich Friedman

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

Original entry on oeis.org

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

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.

A378930 a(n) = n * d(n) / gcd(n, d(n))^2, where d = A000005.

Original entry on oeis.org

1, 1, 6, 12, 10, 6, 14, 2, 3, 10, 22, 2, 26, 14, 60, 80, 34, 3, 38, 30, 84, 22, 46, 3, 75, 26, 108, 42, 58, 60, 62, 48, 132, 34, 140, 4, 74, 38, 156, 5, 82, 84, 86, 66, 30, 46, 94, 120, 147, 75, 204, 78, 106, 108, 220, 7, 228, 58, 118, 5, 122, 62, 42, 448, 260

Offset: 1

Viliam Furík, Dec 11 2024

nonn

Viliam Furík, Table of n, a(n) for n = 1..10000

Cf. A000005, A009191, A009230, A038040.

a:= n-> (d-> n*d/igcd(n,d)^2)(numtheory[tau](n)):
seq(a(n), n=1..65);  # Alois P. Heinz, Dec 11 2024

a[n_]:= n * DivisorSigma[0,n]/GCD[n, DivisorSigma[0,n]]^2; Array[a,65] (* Stefano Spezia, Dec 11 2024 *)

a(n) = my(d = numdiv(n)); n * d / gcd(n, d)^2; \\ Amiram Eldar, Dec 11 2024

a(n) = A038040(n)/A009191(n)^2.

a(n) = A009230(n)/A009191(n).

A306667 Numbers m such that lcm(tau(m), m) = sigma(m) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 6, 32760, 51001180160, 54530444405217553992377326508106948362108928, 133821156044600922812153118065015159487725568, 42274041475824304453686528060845522019324411248640, 48949643430560436794021629524876790263031553747866371344635527168

Offset: 1

Jaroslav Krizek, Mar 04 2019

Numbers m such that A009230(m) = A000203(m).

Subsequence of multiply-perfect numbers (A007691).

			6 is a term because lcm(tau(6), 6) = lcm(4, 6) = 12 = sigma(6).

Giovanni Resta, Table of n, a(n) for n = 1..11 (from A007691 data)

Cf. A000005, A000203, A009230.

Cf. A069810 (gcd(k, sigma(k)) = tau(k)).

[n: n in [1..100000] | LCM(NumberOfDivisors(n), n) eq SumOfDivisors(n)]

a(4)-a(8) computed from A007691 data by Giovanni Resta, Mar 05 2019

A334782 a(n) = Sum_{d|n} lcm(d, tau(d)).

Original entry on oeis.org

1, 3, 7, 15, 11, 21, 15, 23, 16, 33, 23, 45, 27, 45, 77, 103, 35, 48, 39, 105, 105, 69, 47, 77, 86, 81, 124, 141, 59, 231, 63, 199, 161, 105, 165, 108, 75, 117, 189, 153, 83, 315, 87, 213, 176, 141, 95, 397, 162, 258, 245, 249, 107, 372, 253, 205, 273, 177

Offset: 1

Jaroslav Krizek, May 10 2020

nonn

			a(6) = lcm(1, tau(1)) + lcm(2, tau(2)) + lcm(3, tau(3)) + lcm(6, tau(6)) = lcm(1, 1) + lcm(2, 2) + lcm(3, 2) + lcm(6, 4) = 1 + 2 + 6 + 12 = 21.

Cf. A322979 (Sum_{d|n} gcd(d, tau(d))), A334783 (Sum_{d|n} lcm(d, sigma(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009230 (lcm(n, tau(n))).

[&+[LCM(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]

a[n_] := DivisorSum[n, LCM[#, DivisorSigma[0, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)

a(n) = sumdiv(n, d, lcm(d, numdiv(d))); \\ Michel Marcus, May 10 2020

a(p) = 2p + 1 for p = odd primes (A065091).

A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

Original entry on oeis.org

135, 546, 672, 9585, 24570, 51510, 63855, 190008, 251370, 323730, 372438, 486180, 510570, 723550, 819000, 1058910, 1282365, 1284192, 1356030, 3506390, 5416200, 5604480, 6298625, 15593760, 17813250, 18009000, 20740590, 26759370, 27027000, 27081000, 29795040

Offset: 1

Marius A. Burtea, Oct 23 2019

nonn

Composite numbers k verifying equation A009230(k) + A009262(k) = A009242(k).
For any prime number p >= 3 the equality lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)) is satisfied.
The sequence terms are the composite numbers for which the equality is true.

			For k = 135 = 3^3 * 5, tau(k) = 4 * 2 = 2^3, phi(k) = 2 * 3^2 * 4 = 2^3 * 3^2 , sigma(k) = 2^4 * 3 * 5, lcm(k, tau(k)) + lcm(k, phi(k)) =  2^3 * 3^3 * 5 + 2^3 * 3^3 * 5 = 2^4 * 3^3 * 5 and lcm(k, sigma(k)) = lcm(3^3 * 5, 2^4 * 3 * 5) = 2^4 * 3^3 * 5.

Cf. A000005, A000010, A000203, A009242, A009230, A009262, A326416.

[k: k in [1..6000000]| not IsPrime(k) and Lcm(k,NumberOfDivisors(k))+Lcm(k,EulerPhi(k)) eq Lcm(k,SumOfDivisors(k))];

aQ[n_] := CompositeQ[n] && LCM[n, EulerPhi[n]] + LCM[n, DivisorSigma[0, n]] == LCM[n, DivisorSigma[1, n]]; Select[Range[3*10^6], aQ] (* Amiram Eldar, Oct 23 2019 *)

isok(k) = !isprime(k) && (lcm(k, numdiv(k)) + lcm(k, eulerphi(k)) == lcm(k, sigma(k))); \\ Michel Marcus, Oct 24 2019

A334795 a(n) = Product_{d|n} lcm(d, tau(d)) where tau(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 6, 24, 10, 144, 14, 192, 54, 400, 22, 20736, 26, 784, 3600, 15360, 34, 23328, 38, 288000, 7056, 1936, 46, 3981312, 750, 2704, 5832, 790272, 58, 207360000, 62, 1474560, 17424, 4624, 19600, 120932352, 74, 5776, 24336, 92160000, 82, 796594176, 86, 3066624

Offset: 1

Jaroslav Krizek, May 12 2020

From Robert Israel, Jun 25 2020: (Start)
If p is an odd prime, a(p) = 2*p.
If p is a prime > 3, a(p^2) = 6*p^3.
If p and q are distinct odd primes, a(p*q) = 16*p^2*q^2. (End)

			a(6) = lcm(1, tau(1)) * lcm(2, tau(2)) * lcm(3, tau(3)) * lcm(6, tau(6)) = lcm(1, 1) * lcm(2, 2) * lcm(3, 2) * lcm(6, 4) = 1 * 2 * 6 * 12 = 144.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A334782 (Sum_{d|n} lcm(d, tau(d))), A334664 (Product_{d|n} gcd(d, tau(d))).

Cf. A000005 (tau(n)), A009230 (lcm(n, tau(n))).

[&*[LCM(d, #Divisors(d)): d in Divisors(n)]: n in [1..100]]

g:= d -> ilcm(d, numtheory:-tau(d)):
f:= n -> mul(g(d), d = numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Jun 25 2020

a[n_] := Product[LCM[d, DivisorSigma[0, d]], {d, Divisors[n]}]; Array[a, 100] (* Amiram Eldar, May 12 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], numdiv(d[k]))); \\ Michel Marcus, May 12 2020

a(p) = 2p for p = odd primes (A065091).

A258468 a(n) = lcm(n, n - tau(n)).

Original entry on oeis.org

0, 0, 3, 4, 15, 6, 35, 8, 18, 30, 99, 12, 143, 70, 165, 176, 255, 36, 323, 140, 357, 198, 483, 48, 550, 286, 621, 308, 783, 330, 899, 416, 957, 510, 1085, 108, 1295, 646, 1365, 160, 1599, 714, 1763, 836, 585, 966, 2115, 912, 2254, 1100, 2397, 1196

Offset: 1

Carlos Eduardo Olivieri, May 31 2015

For tau see A000005.

			a(5) = 15, since tau(5) = 2, lcm(5, 3) = 15.
a(7) = 35, since tau(7) = 2, lcm(7, 5) = 35.
a(10) = 30, since tau(10) = 4, lcm (10, 6) = 30.

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, A009230, A005563.

Table[LCM[n, n - DivisorSigma[0, n]], {n, 200}]

vector(100, n, lcm(n, n-numdiv(n))) \\ Michel Marcus, May 31 2015

a(n) = lcm(n, n - tau(n)).

a(n) = n * (n - 2) = A005563(n-2) if n is prime.

Edited by Wolfdieter Lang, Jun 16 2015

cp's OEIS Frontend

A033950 Refactorable numbers: number of divisors of k divides k. Also known as tau numbers.

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

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

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A378930 a(n) = n * d(n) / gcd(n, d(n))^2, where d = A000005.

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A306667 Numbers m such that lcm(tau(m), m) = sigma(m) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

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A334782 a(n) = Sum_{d|n} lcm(d, tau(d)).

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A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

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