A051279 -id:A051279 - 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.

A051278 Numbers n such that n = k/d(k) has a unique solution, where d(k) = number of divisors of k.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 20, 21, 22, 26, 32, 33, 34, 35, 36, 38, 39, 42, 46, 50, 51, 55, 57, 58, 60, 62, 65, 66, 69, 70, 74, 75, 77, 78, 82, 85, 86, 87, 90, 91, 93, 94, 95, 96, 98, 100, 102, 106, 108, 110, 111, 114, 115, 118, 119, 122, 123, 126, 128, 129, 130

Offset: 1

N. J. A. Sloane, R. K. Guy

Because d(k) <= 2*sqrt(k), it suffices to check k from 1 to 4*n^2. - Nathaniel Johnston, May 04 2011

A051521(a(n)) = 1. - Reinhard Zumkeller, Dec 28 2011

			36 is the unique number k with k/d(k)=4.

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

Cf. A033950, A036763, A051279, A051280, A051346.

a051278 n = a051278_list !! (n-1)
a051278_list = filter ((== 1) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A051278 := proc(n) local ct,k: ct:=0: for k from 1 to 4*n^2 do if(n=k/tau(k))then ct:=ct+1: fi: od: if(ct=1)then return n: else return NULL: fi: end: seq(A051278(n),n=1..40);

cnt[n_] := Count[Table[n == k/DivisorSigma[0, k], {k, 1, 4*n^2}], True]; Select[Range[130], cnt[#] == 1 &]  (* Jean-François Alcover, Oct 22 2012 *)

A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

3, 25, 40, 49, 54, 121, 125, 135, 140, 169, 189, 216, 220, 250, 260, 289, 297, 340, 351, 361, 375, 380, 400, 459, 460, 500, 513, 529, 580, 620, 621, 675, 729, 740, 770, 783, 820, 837, 841, 860, 875, 882, 910, 940, 961, 999, 1060, 1107, 1152, 1161, 1180, 1188, 1190

Offset: 1

N. J. A. Sloane, R. K. Guy, David W. Wilson

Many terms are of the form a(k) * p^m/(m+1), where p is coprime to the three solutions for k. The sequence of "primitive" terms (i.e. not expressible this way) begins 3, 40, 54, 125, 135, 216, 250.... Are there any such numbers that admit a fourth solution? - Charlie Neder, Feb 13 2019

			There are exactly 3 numbers k, 9, 18 and 24, with k/d(k) = 3.

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

Cf. A033950, A036763, A051278, A051279, A051346.

(Select[Table[k / Length @ Divisors[k], {k, 1, 200000}], IntegerQ] // Sort // Split // Select[#, Length[#] == 3 &] &)[[All, 1]][[1 ;; 53]] (* Jean-François Alcover, Apr 22 2011 *)

A051346 Numbers that can be written as k/d(k) in four or more ways, where d(k) = number of divisors of k.

Original entry on oeis.org

11264, 14175, 28160, 44100, 46464, 51200, 64000, 82944, 95744, 96000, 107008, 109375, 109760, 116160, 129536, 151263, 162624, 163328, 174592, 192000, 208384, 224000, 230912, 239360, 242176, 242550, 246960, 264704, 267520, 281600, 286650, 290304, 298496, 302016

Offset: 1

David W. Wilson

nonn

			From _Jon E. Schoenfield_, Feb 18 2021: (Start)
11264 is a term because it can be written as k/d(k) in four ways:
k =  360448:  360448/d(360448)  =  360448/32 = 11264;
k =  585728:  585728/d(585728)  =  585728/52 = 11264;
k =  630784:  630784/d(630784)  =  630784/56 = 11264;
k = 1115136: 1115136/d(1115136) = 1115136/99 = 11264. (End)

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000005, A051278, A051279, A051280, A217125, A217126, A217127.

(* Assuming 3*10^5 <= k <= 3*10^8 *) ClearAll[cnt]; cnt[] = 0; Do[ If[IntegerQ[n = k/DivisorSigma[0, k]], cnt[n]++; If[cnt[n] >= 4, Print[{n, k, cnt[n]}]]], {k, 3*10^5, 3*10^8}]; Select[Range[310000], cnt[#] >= 4 &] (* _Jean-François Alcover, Sep 28 2012 *)

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

A217125 Numbers n such that n = k/d(k) has exactly 4 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

11264, 14175, 28160, 44100, 46464, 51200, 95744, 96000, 107008, 109375, 109760, 116160, 129536, 151263, 162624, 163328, 174592, 192000, 208384, 224000, 230912, 239360, 242176, 242550, 246960, 264704, 267520, 281600, 286650, 298496, 302016, 323840, 332288

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 11264 for exactly 4 k values: 360448, 585728, 630784 and 1115136.

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A051278, A051279, A051280, A051346, A217126, A217127.

(* Assuming 3*10^5 <= k <= 3*10^8 *) ClearAll[cnt]; cnt[] = 0; Do[ If[IntegerQ[n = k/DivisorSigma[0, k]], cnt[n]++; If[cnt[n] >= 4, Print[{n, k, cnt[n]}]]], {k, 3*10^5, 3*10^8}]; Select[Range[350000], cnt[#] == 4 &] (* _Jean-François Alcover, Sep 28 2012 *)

A217126 Numbers n such that n = k/d(k) has exactly 5 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

64000, 290304, 352000, 432000, 544000, 608000, 736000, 928000, 992000, 1036800, 1184000, 1312000, 1376000, 1504000, 1512000, 1596672, 1696000, 1888000, 1952000, 2100875, 2144000, 2272000, 2336000, 2467584, 2515968, 2528000, 2592000, 2656000, 2757888, 2848000

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 64000 for exactly 5 k values: 4096000, 6656000, 7040000, 7168000 and 11520000.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A051278, A051279, A051280, A051346, A217125, A217127.

A217127 Numbers n such that n = k/d(k) has exactly 6 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

82944, 456192, 705024, 787968, 953856, 1202688, 1285632, 1534464, 1700352, 1783296, 1949184, 2198016, 2446848, 2529792, 2778624, 2944512, 3027456, 3276288, 3345408, 3442176, 3691008, 3877632, 4022784, 4188672, 4271616, 4333824, 4437504, 4520448, 4686336

Offset: 1

Donovan Johnson, Sep 27 2012

nonn

			k/d(k) = 82944 for exactly 6 k values: 7962624, 12939264, 13271040, 13934592, 21565440 and 23224320.

Donovan Johnson, Table of n, a(n) for n = 1..300

Cf. A051278, A051279, A051280, A051346, A217125, A217126.

A051522 Least positive integer with exactly n representations as k/d(k), where d(k) = number of divisors of k.

Original entry on oeis.org

18, 4, 1, 3, 11264, 64000, 82944, 27599616

Offset: 0

David W. Wilson

27599616 is the only number < 4*10^7 with exactly 7 representations as k/d(k). - Donovan Johnson, Sep 29 2012

			Contribution from _Donovan Johnson_, Sep 29 2012: (Start)
a(1) = A051278(1) = 4. k/d(k) = 4 for exactly 1 k value: 36.
a(2) = A051279(1) = 1. k/d(k) = 1 for exactly 2 k values: 1 and 2.
a(3) = A051280(1) = 3. k/d(k) = 3 for exactly 3 k values: 9, 18 and 24.
a(4) = A217125(1) = 11264. k/d(k) = 11264 for exactly 4 k values: 360448, 585728, 630784 and 1115136.
a(5) = A217126(1) = 64000. k/d(k) = 64000 for exactly 5 k values: 4096000, 6656000, 7040000, 7168000 and 11520000.
a(6) = A217127(1) = 82944. k/d(k) = 82944 for exactly 6 k values: 7962624, 12939264, 13271040, 13934592, 21565440 and 23224320.
a(7) = 27599616. k/d(k) = 27599616 for exactly 7 k values: 10598252544, 17222160384, 17663754240, 18215746560, 18546941952, 28703600640 and 30911569920. (End)

Cf. A051278, A051279, A051280, A217125, A217126, A217127.

a(4)-a(6) verified and a(7) added by Donovan Johnson, Sep 29 2012

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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A051278 Numbers n such that n = k/d(k) has a unique solution, where d(k) = number of divisors of k.

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A051280 Numbers n such that n = k/d(k) has exactly 3 solutions, where d(k) = number of divisors of k.

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A051346 Numbers that can be written as k/d(k) in four or more ways, where d(k) = number of divisors of k.

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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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A217125 Numbers n such that n = k/d(k) has exactly 4 solutions, where d(k) = number of divisors of k.

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