A033950 -id:A033950 - cp's OEIS frontend

A036878 a(n) = p^(p-1) where p = prime(n).

2, 9, 625, 117649, 25937424601, 23298085122481, 48661191875666868481, 104127350297911241532841, 907846434775996175406740561329, 88540901833145211536614766025207452637361, 550618520345910837374536871905139185678862401

Offset: 1

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

Also the least refactorable number (A033950) that has the n-th prime as its least prime factor. - Robert G. Wilson v, Jun 28 2006

			5^(5-1) = 5^4 = 625.

Vincenzo Librandi, Table of n, a(n) for n = 1..77
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

These integers are refactorable -- i.e., the number of divisors divides the number itself, cf. A033950.
Subset of A062981. Subsequence of A000169.
Subsequence of A111134 and A246655.

[p^(p-1): p in PrimesUpTo(50)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime@n^(Prime@n - 1), {n, 10}] (* Robert G. Wilson v, Jun 28 2006 *)
#^(#-1)&/@Prime[Range[10]] (* Harvey P. Dale, Oct 23 2015 *)

a(n) = my(p=prime(n)); p^(p-1); \\ Michel Marcus, Jan 24 2019

A036896 Odd refactorable numbers.

Original entry on oeis.org

1, 9, 225, 441, 625, 1089, 1521, 2025, 2601, 3249, 4761, 5625, 6561, 7569, 8649, 12321, 15129, 16641, 19881, 25281, 31329, 33489, 35721, 40401, 45369, 47961, 50625, 56169, 62001, 71289, 84681, 91809, 95481, 99225, 103041, 106929, 114921

Offset: 1

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

Odd refactorable numbers are always squares.

All terms = 1 (mod 8). [Zak Seidov, May 25 2010]

			9 is refactorable because tau(9)=3 and 3 divides 9.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1001 terms from Harvey P. Dale)
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics

Subsequence of A033950 and of A016754.

Do[If[IntegerQ[n/DivisorSigma[0, n]], Print[n]], {n, 1, 100000, 2}]
Select[Range[1,1001,2]^2,Divisible[#,DivisorSigma[0,#]]&] (* Harvey P. Dale, Jan 22 2012 *)

is(n)=n%2&&issquare(n)&&n%numdiv(n)==0 \\ Charles R Greathouse IV, Apr 23 2013

list(lim)=my(v=List(),f);forstep(n=1,sqrtint(lim\1),2,f=factor(n)[,2];if(n^2%prod(i=1,#f,2*f[i]+1)==0,listput(v,n^2))); Vec(v) \\ Charles R Greathouse IV, Apr 23 2013

from itertools import count, islice
from math import prod
from sympy import factorint
def A036896_gen(): # generator of terms
    for n in count(1,2):
        if not (m:=n**2)%prod(e<<1|1 for e in factorint(n).values()): yield m
A036896_list = list(islice(A036896_gen(),40)) # Chai Wah Wu, Oct 04 2024

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 *)

A051279 Numbers n such that n = k/d(k) has exactly 2 solutions, where d(k) = number of divisors of k.

Original entry on oeis.org

1, 2, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 28, 29, 31, 37, 41, 43, 44, 47, 48, 52, 53, 56, 59, 61, 67, 68, 71, 73, 76, 79, 80, 81, 83, 84, 88, 89, 92, 97, 101, 103, 104, 107, 109, 113, 116, 120, 124, 127, 131, 132, 136, 137, 139, 148, 149, 151, 152, 154, 156

Offset: 1

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

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)) = 2. - Reinhard Zumkeller, Dec 28 2011

			There are exactly 2 numbers k, 40 and 60, with k/d(k)=5.

Nathaniel Johnston and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 150 terms from Nathaniel Johnston)

Cf. A033950, A036763, A051278, A051280, A051346.

a051279 n = a051279_list !! (n-1)
a051279_list = filter ((== 2) . a051521) [1..]
-- Reinhard Zumkeller, Dec 28 2011

with(numtheory): A051279 := 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=2)then return n: else return NULL: fi: end: seq(A051279(n), n=1..40); # Nathaniel Johnston, May 04 2011

A051279 = Reap[Do[ct = 0; For[k = 1, k <= 4*n^2, k++, If[n == k/DivisorSigma[0, k], ct++]]; If[ct == 2, Print[n]; Sow[n]], {n, 1, 160}]][[2, 1]](* Jean-François Alcover, Apr 16 2012, after Nathaniel Johnston *)

A036898 List of pairs of consecutive refactorable numbers.

Original entry on oeis.org

1, 2, 8, 9, 1520, 1521, 50624, 50625, 62000, 62001, 103040, 103041, 199808, 199809, 221840, 221841, 269360, 269361, 463760, 463761, 690560, 690561, 848240, 848241, 986048, 986049, 1252160, 1252161, 1418480, 1418481, 2169728, 2169729, 2692880

Offset: 1

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

nonn

Zelinsky (2002, Theorem 59, p. 15) proved that if k > 1, k and k+1 are both refactorable numbers, then k is even. As a result, a(n) == n-1 (mod 2) for n >= 3. See also A114617. - Jianing Song, Apr 01 2021

			8 is refactorable because tau(8)=4 and 4 divides 8.
9 is refactorable because tau(9)=3 and 3 divides 9.

Donovan Johnson, Table of n, a(n) for n = 1..2000
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. Colton, HR - Automatic Theory Formation in Pure Mathematics
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. A033950, A114617.

SequencePosition[Table[If[Divisible[n,DivisorSigma[0,n]],1,0],{n,27*10^5}],{1,1}]//Flatten (* Harvey P. Dale, Dec 07 2021 *)

isrefac(n) = ! (n % numdiv(n));
lista(nn) = {for (n = 1, nn, if (isrefac(n) && isrefac(n+1), print1(n, ", ", n+1, ", ")););} \\ Michel Marcus, Aug 31 2013

A048166 Numbers k that are divisible by the number of unitary divisors of k (A034444).

Original entry on oeis.org

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 64, 68, 72, 76, 80, 88, 92, 96, 100, 104, 108, 112, 116, 120, 124, 128, 136, 144, 148, 152, 160, 164, 168, 172, 176, 184, 188, 192, 196, 200, 208, 212, 216, 224, 232, 236, 240, 244, 248, 256, 264, 268

Offset: 1

Labos Elemer

nonn

			a(81) = 392 = 2^3*7^2 has 4 unitary divisors, {1, 392, 8, 49}, and 4 divides 392.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms n = 1..5000 from G. C. Greubel)

Cf. A033950, A034444, A000005, A048167, A048481.

okQ[n_] := Divisible[n, DivisorSum[n, Boole[CoprimeQ[#, n/#]]&]]; Select[ Range[300], okQ] (* Jean-François Alcover, Dec 05 2015 *)
Select[Range[270], Divisible[#, 2^PrimeNu[#]] &] (* Amiram Eldar, Jul 16 2019 *)

isok(n) = !(n % sumdiv(n, d, gcd(d, n/d)==1)); \\ Michel Marcus, Feb 25 2014

isok(n) = !(n % 2^omega(n)); \\ Amiram Eldar, Jul 16 2019

Binomial transform of [1, 1, 1, 1, -3, 5, -7, 9, -11, 13, ...]. Binomial transform of this sequence is A048481. - Gary W. Adamson, Oct 23 2007

A049439 Numbers k such that the number of odd divisors of k is an odd divisor of k.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 128, 144, 225, 256, 288, 441, 450, 512, 576, 625, 882, 900, 1024, 1089, 1152, 1250, 1521, 1764, 1800, 2025, 2048, 2178, 2304, 2500, 2601, 3042, 3249, 3528, 3600, 4050, 4096, 4356, 4608, 4761, 5000, 5202, 5625, 6084

Offset: 1

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

Invented by the HR concept formation program.

Sequence consists of all numbers of the form A000079(k)*A036896(m). - Matthew Vandermast, Nov 14 2010

			There are 3 odd divisors of 18, namely 1,3 and 9 and 3 itself is an odd divisor of 18.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.

Contains A000079 and A036896.
Cf. A001227, A033950.
Subsequence of A028982. Includes A120349, A120358, A120359, A120361, A181795. See also A181794.

a049439 n = a049439_list !! (n-1)
a049439_list = filter (\x -> ((length $ oddDivs x) `elem` oddDivs x)) [1..]
   where oddDivs n = [d | d <- [1,3..n], mod n d == 0]
-- Reinhard Zumkeller, Aug 17 2011

ok[n_] := (d = Length @ Select[Divisors[n], OddQ] ;
  IntegerQ[n/d] && OddQ[d]); Select[Range[6100], ok]
(* Jean-François Alcover, Apr 22 2011 *)
odQ[n_]:=Module[{ods=Select[Divisors[n],OddQ]},MemberQ[ods,Length[ ods]]]; Select[Range[7000],odQ] (* Harvey P. Dale, Dec 18 2011 *)
Select[Range[6000], OddQ[(d = DivisorSigma[0, #/2^IntegerExponent[#, 2]])] && Divisible[#, d] &] (* Amiram Eldar, Jun 12 2022 *)

is(n)=my(d=numdiv(n>>valuation(n,2))); d%2 && n%d==0 \\ Charles R Greathouse IV, Feb 07 2017

a(n) = A000079(k)*A016754(m) for appropriate k, m. - Reinhard Zumkeller, Jun 05 2008

Example corrected by Harvey P. Dale, Jul 14 2011

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 *)

A057265 Even refactorable numbers (i.e., the number of divisors is itself a divisor and it is also even).

Original entry on oeis.org

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

Offset: 1

Simon Colton (simonco(AT)cs.york.ac.uk), Aug 21 2000

Invented by the HR mathematical theory formation program.

			18 is refactorable because tau(18) = 6 and 6 divides 18 and 18 is even.

S. Colton, Unpublished PhD Thesis, University of Edinburgh, 2000

Amiram Eldar, Table of n, a(n) for n = 1..10000
Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
Simon Colton, HR - Automatic Theory Formation in Pure Mathematics

Cf. A033950, A036896.

[k:k in [2..500 by 2]| IsIntegral(k/d) and IsEven(d) where d is #Divisors(k)]; // Marius A. Burtea, Jan 13 2020

rfnQ[n_]:=Module[{ds=DivisorSigma[0,n]},Divisible[n,ds] && EvenQ[ds]];Select[Range[2,500,2],rfnQ]  (* Harvey P. Dale, Mar 14 2011 *)

Corrected (erroneous term 36 removed) by Harvey P. Dale, Mar 14 2011

A073904 Smallest multiple k*n of n having n divisors.

Original entry on oeis.org

1, 2, 9, 8, 625, 12, 117649, 24, 36, 80, 25937424601, 60, 23298085122481, 448, 2025, 384, 48661191875666868481, 180, 104127350297911241532841, 240, 35721, 11264, 907846434775996175406740561329, 360, 10000, 53248, 26244, 1344

Offset: 1

Amarnath Murthy, Aug 18 2002

Smallest refactorable number, m, such that m=k*n has n divisors. - Robert G. Wilson v, Oct 31 2005

			Smallest multiple a(n)=k*n; a(1)=1*1, a(2)=1*2, a(3)=3*3, a(4)=2*4, a(5)=125*5, a(6)=2*6, ... having d(k*n)=n divisors; d(1)=1, d(2)=2, d(3^2)=3, d(2^3)=4, d(5^4)=5, d(2^2*3)=3*2=6, ...

Jon E. Schoenfield, Table of n, a(n) for n = 1..388 (first 115 terms from Carole Dubois)
Carole Dubois, semi-Logarithmic pin plot of A073904(n)

Cf. A076931, A050376, A005179, A037992, A050376, A111172, A299795.
Cf. A033950 (refactorable numbers, also known as tau numbers).
Cf. A110821 (SuperRefactorable numbers).

f[n_] := Block[{k = 1}, If[ PrimeQ[n], n^(n - 1), While[d = DivisorSigma[0, k*n]; d != n, k++ ]; k*n]]; Table[ f[n], {n, 28}] (* Robert G. Wilson v *)

If p is a prime then a(p) = p^(p-1). If n = p^2 then a(n) = 2^(p-1)*p^(p-1).
a(p^r) = (2*3*5*...*p_r)^(p-1) for r < p <= p_r. a(p^r) = (2*3*...*p_(r-1))^(p-1)*p^(p-1) for p > p_r. Else a(p^r) = ...? for r >= p. Problem a(2^r) = ...? Cf. A005179(p^n)=(2*3*...*p_n)^(p-1) for p_n < 2^p. - Thomas Ordowski, Aug 20 2005
a(p^r) = (2*3...*p_(r-1)*p)^(p-1) for p > p_r; else a(p^r) = (2*3...*p...*p_m)^(p-1)*p^(p^k-p) for p <= p_r and p_m < 2^p, where m=r-k+1 for smallest k such that p^k > r, so k=floor(log(r)/log(p))+1 and p > log(p_m)/log(2). Examples: If k=1 then a(p^r) = (2*3*...*p_r)^(p-1) for r < p <= p_r. If p=2 then a(2^r) = (2*3*...*p_m)*2^(2^k-2) for r < 5. For instance, let r=4 so k=3, m=2 and a(2^4)=384. - Thomas Ordowski, Aug 22 2005
If p is a prime and n=p^r then a(p^r) = (s_1*s_2*...*s_r)^(p-1) where (s_r) is a permutation of the (ascending sequence) numbers of the form q^(p^j) for every prime q and j>=0; permutation such that s_(p^j)=p^(p^j) and shifted remainder. For example, if p=3 then (s_r): 3, 2, 3^3, 5, 7, 2^3, 11, 13, 3^9, 17, 19, ... so a(3^r) = (3*2*27*5*...*s_r)^2. - Thomas Ordowski, Aug 29 2005
If n=2^r then a(2^r) is the product of the first r members of the A109429 sequence. - Thomas Ordowski, Aug 29 2005
a(n) = n * A076931(n). - Thomas Ordowski, Oct 07 2005
a(4) = 8; a(2*prime(n)) = A299795(n), for n>1. - Bernard Schott, Nov 06 2022

a(12) corrected by Thomas Ordowski, Aug 18 2005
Further corrections from Thomas Ordowski, Oct 07 2005
a(21), a(27) & a(28) from Robert G. Wilson v, Oct 31 2005

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A036878 a(n) = p^(p-1) where p = prime(n).

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A036896 Odd refactorable numbers.

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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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A051279 Numbers n such that n = k/d(k) has exactly 2 solutions, where d(k) = number of divisors of k.

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A036898 List of pairs of consecutive refactorable numbers.

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A048166 Numbers k that are divisible by the number of unitary divisors of k (A034444).

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A049439 Numbers k such that the number of odd divisors of k is an odd divisor of k.

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