A036762 -id:A036762 - 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

A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

Original entry on oeis.org

18, 27, 30, 45, 63, 64, 72, 99, 105, 112, 117, 144, 153, 160, 162, 165, 171, 195, 207, 225, 243, 252, 255, 261, 279, 285, 288, 294, 320, 333, 336, 345, 352, 360, 369, 387, 396, 405, 416, 423, 435, 441, 465, 468, 477, 490, 504, 531, 544, 549, 555, 567, 576

Offset: 1

Labos Elemer

nonn

A special case of a bound on d(n) by Erdős and Suranyi (1960) was used to get a limit: a = x/d(x) > 0.5*sqrt(x) and below 4194304 a computer test shows these values did not occur as x = a*d(x). For larger x this is impossible since if d(x) < sqrt(x), then x/d(x) > sqrt(4194304) = 2048 > the given terms.
A051521(a(n)) = 0. - Reinhard Zumkeller, Dec 28 2011
This sequence contains all numbers of the form k = 9p, p prime (i.e., k = 18, 27, 45, 63, 99, ...). - Jianing Song, Nov 25 2018

			No natural number x exists for which x = 18*d(x), so 18 is a term.

P. Erdős and J. Suranyi, Selected Topics in Number Theory, Tankonyvkiado, Budapest, 1960 (in Hungarian).
P. Erdős and J. Suranyi, Selected Topics in Number Theory, Springer, New York, 2003 (in English).

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A000005, A033950, A036761, A036762, A036764, A051278, A051279, A051280, A051521.

a036763 n = a036763_list !! (n-1)
a036763_list = filter ((== 0) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A036763 := proc(n) local k,p: for k from 1 to 4*n^2 do p:=n*k: if(p=n*tau(p))then return NULL: fi: od: return n: end: seq(A036763(n),n=1..100); # Nathaniel Johnston, May 04 2011

noSolQ[n_] := !AnyTrue[Range[4*n^2], # == DivisorSigma[0, n*#]& ];
Reap[Do[If[noSolQ[n], Print[n]; Sow[n]], {n, 600}]][[2, 1]] (* Jean-François Alcover, Jan 30 2018 *)

Definition corrected by N. J. A. Sloane, May 18 2022 at the suggestion of David James Sycamore.

A039819 Number of divisors of n-th refactorable number (A033950(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 6, 8, 9, 8, 8, 12, 12, 10, 12, 8, 12, 8, 12, 8, 12, 8, 8, 12, 18, 8, 12, 9, 12, 8, 20, 8, 18, 12, 18, 8, 8, 8, 12, 24, 12, 8, 16, 18, 8, 9, 12, 14, 18, 18, 8, 24, 8, 12, 24, 12, 8, 20, 12, 8, 8, 24, 18, 5, 8, 12, 16, 8, 24, 18, 12, 8, 30, 12, 8, 24, 12, 8, 8, 18, 12, 8, 24, 8

Offset: 1

N. J. A. Sloane

nonn

A number n is refactorable if the number of divisors of n divides n.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A033950, A000005, A036762, A281188.

v:=[ n: n in [1..900] | n mod NumberOfDivisors(n) eq 0 ]; [NumberOfDivisors(v[i]): i in [1..#v]]; // Marius A. Burtea, Jul 02 2019

fQ[n_] := Mod[n, DivisorSigma[0, n]] == 0; DivisorSigma[0, # ] & /@ Select[ Range[1000], fQ[ # ] &] (* Robert G. Wilson v *)

A033950(n)/a(n) = A036762(n).

a(n) = A000005(A033950(n)). - Omar E. Pol, Jan 17 2017

More terms from Robert G. Wilson v, Oct 29 2005

Minor edits by Franklin T. Adams-Watters, Jan 17 2017

A051521 Number of ways to represent n as k/d(k), where d(k) = A000005(k) is the number of divisors of k.

Original entry on oeis.org

2, 2, 3, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 0, 2, 1, 1, 1, 2, 2, 3, 1, 0, 2, 2, 0, 2, 1, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 0, 1, 2, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 1, 2, 1, 2, 1, 0, 0, 1, 1, 2, 2, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1

Offset: 1

David W. Wilson

nonn

From Jianing Song, Nov 25 2018: (Start)
a(9p) = 0 for all primes p. Here is a brief proof: a(18) = a(27) = a(45) = a(63) = 0. Now let p be a prime >= 11.
If there is an x such that d(9p*x) = x, let x = p^a*3^b*y, gcd(p, y) = gcd(3, y) = 1, then p^a*3^b*y = d(p^(a+1)*3^(b+2)*y) = (a + 2)*(b + 3)*d(y). Since y >= d(y), we must have (a + 2)*(b + 3) >= p^a*3^b >= 11^a*3^b. If a >= 1, then 3 >= (b + 3)/3^b >= 11^a/(a + 2) >= 11/3, a contradiction. So a = 0. 3^b/(b + 3) <= 2, so b = 0, 1, 2.
Case (i): b = 0, then y = 6*d(y), which has a unique solution y = 72. But gcd(3, 72) != 1, a contradiction,
Case (ii): b = 1, then y = (8/3)*d(y), which has no solution.
Case (iii): b = 2, then y = (10/9)*d(y), which has no solution.
Similarly, it can be proved that a(81p) = 0 for all primes p. (End)

			There are a(1) = 2 numbers k for which k/d(k) = 1, namely k = 1 and k = 2.
There are a(2) = 2 numbers k for which k/d(k) = 2, namely k = 8 and k = 12.
There are a(3) = 3 numbers k for which k/d(k) = 3, namely k = 9, 18 and 24.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000005 (number of divisors), A033950, A036762, A036763 (indices of 0s), A036764, A051278 (indices of 1s), A051279 (indices of 2s).

a051521 n = length [k | k <- [1..4*n^2],
                        let d = a000005 k, divMod k d == (n,0)]
-- Reinhard Zumkeller, Dec 28 2011

a[n_] := Count[Table[n == k/DivisorSigma[0, k], {k, 1, 4*n^2}], True]; Table[a[n], {n, 1, 100}]  (* Jean-François Alcover, Oct 22 2012 *)

a(A036763(n)) = 0; a(A051278(n)) = 1; a(A051279(n)) = 2. - Reinhard Zumkeller, Dec 28 2011

A048167 Integer quotients of numbers k divisible by the number of unitary divisors of k (A034444).

Original entry on oeis.org

1, 1, 2, 4, 3, 8, 5, 6, 7, 16, 9, 10, 11, 12, 13, 14, 32, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 15, 31, 64, 34, 36, 37, 38, 40, 41, 21, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 58, 59, 30, 61, 62, 128, 33, 67, 68, 35, 71, 72, 73, 74, 76, 39, 79, 80, 81, 82, 83, 42

Offset: 1

Labos Elemer

nonn

			1008 = 2*2*2*3*3*7 has 8 unitary divisors {1,1008,7,144,9,112,16,63}, 8 divides 1008 and the quotient is 126, a term in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034444, A036762, A000005, A048166.

f[n_] := n / 2^PrimeNu[n]; f /@ Select[Range[340], IntegerQ[f[#]] &] (* Amiram Eldar, Jul 16 2019 *)

a(n) = A048166(n)/A034444(A048166(n)). - Amiram Eldar, Jul 16 2019

A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.

Original entry on oeis.org

-1, -2, -2, -2, -2, -4, -2, 0, 0, -4, -2, 0, -2, -4, 2, 8, -2, 0, -2, 8, 6, -4, -2, 16, 10, -4, 18, 16, -2, 4, -2, 32, 14, -4, 26, 36, -2, -4, 18, 48, -2, 12, -2, 32, 54, -4, -2, 64, 28, 20, 26, 40, -2, 36, 50, 80, 30, -4, -2, 72, -2, -4, 90, 96, 62, 28, -2, 56, 38, 52, -2, 144, -2, -4, 90, 64, 86, 36, -2, 160, 108, -4, -2, 120, 86, -4

Offset: 1

Labos Elemer, Jul 20 2001

sign

It can be shown that phi(n)*tau(n) >= n, which means that quotient = n/tau(n) <= phi(n); note: a(n)+5 is positive.

The value is always positive except when a(n) = 0 for {8,9,12}; or a(n) = -2 for primes together with 4 (i.e., for A046022 but without 1); or a(n) = -4 for A001747 (without 2 and 4); or a(n) = -1 for n = 1.

Harry J. Smith, Table of n, a(n) for n = 1..2000

Cf. A000010, A000005, A062355, A062516, A046022, A001747, A036762, A036763, A051728, A051729, A051730.

Table[EulerPhi[n]DivisorSigma[0,n]-2n,{n,90}] (* Harvey P. Dale, Feb 03 2021 *)

a(n)={eulerphi(n)*numdiv(n) - 2*n} \\ Harry J. Smith, Aug 11 2009

a(n) = A062355(n) - 2*n. - Amiram Eldar, Jul 10 2024

Offset changed from 0 to 1 by Harry J. Smith, Aug 11 2009

A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

Original entry on oeis.org

1, 8, 9, 72, 13440, 21120, 24960, 29568, 32640, 34944, 36480, 44160, 45696, 49280, 51072, 54912, 55680, 58240, 59520, 61824, 71040, 71808, 76160, 77952, 78720, 80256, 82560, 83328, 84864, 85120, 90240, 91520, 94848, 97152, 99456, 101760, 103040, 110208, 113280, 114816, 115584, 117120, 119680

Offset: 1

Ilya Gutkovskiy, May 12 2018

nonn

Numbers k such that A000005(k)*A007947(k) = k.
Fixed points of A304409.
All terms are refactorable numbers (A033950).

			13440 is a term because 13440 = 2^7*3*5*7 = 2*(7 + 1) * 3*(1 + 1) * 5*(1 + 1) * 7*(1 + 1).

Cf. A000005, A000026, A007947, A008478, A033950, A036762, A048768, A078779, A190473, A304409.

a[n_] := Times @@ (#[[1]] (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Select[Range[120000], a[#] == # &]

isok(k) = {my(f = factor(k)); numdiv(f) * vecprod(f[, 1]) == k;} \\ Amiram Eldar, Jan 31 2025

A341781 Refactorable numbers (or tau numbers, A033950) k such that k/tau(k) is even, where tau(k) = A000005(k).

Original entry on oeis.org

8, 12, 36, 72, 80, 96, 128, 180, 240, 252, 288, 384, 396, 448, 468, 480, 560, 612, 640, 672, 684, 720, 828, 864, 880, 896, 972, 1040, 1044, 1056, 1116, 1152, 1200, 1248, 1332, 1344, 1360, 1408, 1440, 1476, 1520, 1548, 1620, 1632, 1664, 1680, 1692, 1800, 1824

Offset: 1

Amiram Eldar, Feb 19 2021

nonn

Zelinsky (2002) called these numbers p-generators. He proved that these are the tau numbers k such that for any prime p, if p does not divide k then p*k is also a tau number. He used these numbers to prove that the number of tau numbers not exceeding m is > pi(m)/2 for all m > 7.42*10^13, where pi(m) = A000720(m).

			8 is a term since 8/tau(8) = 8/4 = 2 is even.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, A033950, A036762, A039819, A281188.

q[n_] := Divisible[n, (d = DivisorSigma[0, n])] && EvenQ[n/d]; Select[Range[2000], q]

isok(k) = my(q=k/numdiv(k)); (denominator(q)==1) && ((q%2) == 0); \\ Michel Marcus, Feb 20 2021

A366144 a(n) = n/d(n) if d(n) | n, otherwise a(n) = n*d(n), where d(n) = A000005(n) is the number of divisors of n.

Original entry on oeis.org

1, 1, 6, 12, 10, 24, 14, 2, 3, 40, 22, 2, 26, 56, 60, 80, 34, 3, 38, 120, 84, 88, 46, 3, 75, 104, 108, 168, 58, 240, 62, 192, 132, 136, 140, 4, 74, 152, 156, 5, 82, 336, 86, 264, 270, 184, 94, 480, 147, 300, 204, 312, 106, 432, 220, 7, 228, 232, 118, 5, 122, 248, 378, 448, 260, 528, 134

Offset: 1

Neal Gersh Tolunsky, Sep 30 2023

			n=3 has d(3) = 2 divisors (like all primes) and 3 is not divisible by 2, so we multiply: a(3) = 3*2 = 6.
n=8 has d(8) = 4 divisors and 8 is divisible by 4, so we divide: a(8) = 8/4 = 2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^14, showing primes in red, composite prime powers in gold, even squarefree semiprimes in light green, other squarefree composites in dark green, and numbers neither squarefree nor prime powers in blue. Powerful numbers that are not prime powers are highlighted in light blue.
Neal Gersh Tolunsky, Graph of first 150000 terms.

Cf. A000005, A038040, A033950, A036762.

Cf. A366067 (iterate starting 578).

a[n_] := n * If[Divisible[n, d = DivisorSigma[0, n]], 1/d, d]; Array[a, 100] (* Amiram Eldar, Oct 01 2023 *)

a(n) = my(d=numdiv(n)); if (n % d, n*d, n/d); \\ Michel Marcus, Oct 01 2023

from sympy import divisor_count
def A366144(n): return n*d if (q:=divmod(n,d:=int(divisor_count(n))))[1] else q[0] # Chai Wah Wu, Oct 02 2023

sqrt(n)/2 <= a(n) <= 2*n*sqrt(n). - Yifan Xie, Oct 01 2023

A063080 Numbers k such that k/d(k) is prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

8, 9, 12, 18, 24, 40, 56, 60, 84, 88, 104, 132, 136, 152, 156, 184, 204, 228, 232, 248, 276, 296, 328, 344, 348, 372, 376, 424, 444, 472, 488, 492, 516, 536, 564, 568, 584, 632, 636, 664, 708, 712, 732, 776, 804, 808, 824, 852, 856, 872, 876, 904, 948, 996

Offset: 1

Jason Earls, Aug 05 2001

If p is an odd prime, then 8*p is a term. - Amiram Eldar, Apr 17 2024

			k = 18: 18/d(18) = 3 a prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000005, A039819, A036762, A051278, A051279, A051280, A036763, A033950.

Select[Range[1000],PrimeQ[#/DivisorSigma[0,#]]&] (* Harvey P. Dale, Sep 01 2014 *)

ok(m)={my(d=numdiv(m)); m%d==0 && isprime(m/d)}

cp's OEIS Frontend

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

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A036763 Numbers k such that k*d(x) = x has no solution for x, where d (A000005) is the number of divisors; sequence gives impossible x/d(x) quotients in order of magnitude.

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A039819 Number of divisors of n-th refactorable number (A033950(n)).

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A051521 Number of ways to represent n as k/d(k), where d(k) = A000005(k) is the number of divisors of k.

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A048167 Integer quotients of numbers k divisible by the number of unitary divisors of k (A034444).

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A062816 a(n) = phi(n)*tau(n) - 2n = A000010(n)*A000005(n) - 2*n.

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A304410 Numbers k such that k = Product (p_j^e_j) = Product (p_j*(e_j + 1)).

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A062816 a(n) = phi(n)tau(n) - 2n = A000010(n)A000005(n) - 2*n.