A007422 -id:A007422 - cp's OEIS frontend

A007956 Product of the proper divisors of n.

1, 1, 1, 2, 1, 6, 1, 8, 3, 10, 1, 144, 1, 14, 15, 64, 1, 324, 1, 400, 21, 22, 1, 13824, 5, 26, 27, 784, 1, 27000, 1, 1024, 33, 34, 35, 279936, 1, 38, 39, 64000, 1, 74088, 1, 1936, 2025, 46, 1, 5308416, 7, 2500, 51, 2704, 1, 157464, 55, 175616, 57, 58, 1, 777600000, 1, 62, 3969, 32768, 65

Offset: 1

R. Muller

From Bernard Schott, Feb 01 2019: (Start)
a(n) = 1 iff n = 1 or n is prime.
a(n) = n when n > 1 iff n has exactly four divisors, equally, iff n is either the cube of a prime or the product of two different primes, so iff n belongs to A030513 (very nice proof in Sierpiński).
a(p^3) = 1 * p * p^2 = p^3; a(p*q) = 1 * p * q = p*q.
As a(1) = 1, {1} Union A030513 = A007422, fixed points of this sequence. (End)

			a(18) = 1 * 2 * 3 * 6 * 9 = 324. - _Bernard Schott_, Jan 31 2019

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 57.
Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Zhu Weiyi, On the divisor product sequences, Smarandache Notions J., Vol. 14 (2004), pp. 144-146.

Cf. A000005, A000720, A007955, A048671, A182936, A027751, A066729.
Cf. A007422 (fixed points). A030513 (subsequence).
Cf. A001065 (sums of proper divisors).

a007956 = product . a027751_row
-- Reinhard Zumkeller, Feb 04 2013, Nov 02 2011

A007956 := n -> mul(i,i=op(numtheory[divisors](n) minus {1,n}));
seq(A007956(i), i=1..79); # Peter Luschny, Mar 22 2011

Table[Times@@Most[Divisors[n]], {n, 65}] (* Alonso del Arte, Apr 18 2011 *)
a[n_] := n^(DivisorSigma[0, n]/2 - 1); Table[a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 07 2013 *)

A007956(n) = local(a);a=1;fordiv(n,d,a=a*d);a/n \\ Michael B. Porter, Dec 01 2009

a(n)=my(k); if(issquare(n, &k), k^(numdiv(n)-2), n^(numdiv(n)/2-1)) \\ Charles R Greathouse IV, Oct 15 2015

from math import isqrt
from sympy import divisor_count
def A007956(n): return isqrt(n)**(d-2) if (d:=divisor_count(n))&1 else n**((d>>1)-1) # Chai Wah Wu, Jun 18 2023

a(n) = A007955(n)/n = n^(A000005(n)/2-1) = sqrt(n^(number of factors of n other than 1 and n)).
a(n) = Product_{k=1..A000005(n)-1} A027751(n,k). - Reinhard Zumkeller, Feb 04 2013
a(n) = A240694(n, A000005(n)-1) for n > 1. - Reinhard Zumkeller, Apr 10 2014
Sum_{k=1..n} 1/a(k) ~ pi(n) + log(log(n))^2 + c_1*log(log(n)) + c_2 + O(log(log(n))/log(n)), where pi(n) = A000720(n) and c_1 and c_2 are constants (Weiyi, 2004; Sandor and Crstici, 2004). - Amiram Eldar, Oct 29 2022

More terms from Scott Lindhurst (ScottL(AT)alumni.princeton.edu)

A030513 Numbers with 4 divisors.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187

Offset: 1

Jeff Burch

Essentially the same as A007422.
Numbers which are either the product of two distinct primes (A006881) or the cube of a prime (A030078).
4*a(n) are the solutions to A048272(x) = Sum_{d|x} (-1)^d = 4. - Benoit Cloitre, Apr 14 2002
Since A119479(4)=3, there are never more than 3 consecutive integers in the sequence. Triples of consecutive integers start at 33, 85, 93, 141, 201, ... (A039833). No such triple contains a term of the form p^3. - Ivan Neretin, Feb 08 2016
Numbers that are equal to the product of their proper divisors (A007956) (proof in Sierpiński). - Bernard Schott, Apr 04 2022

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 2 p. 174, Warsaw, 1964.

Cf. A000005, A007422, A007956, A030515, A035533, A048272, A119479.

Equals the disjoint union of A006881 and A030078.

[n: n in [1..200] | DivisorSigma(0, n) eq 4]; // Vincenzo Librandi, Jul 16 2015

Select[Range[200], DivisorSigma[0,#]==4&] (* Harvey P. Dale, Apr 06 2011 *)

is(n)=numdiv(n)==4 \\ Charles R Greathouse IV, May 18 2015

from math import isqrt
from sympy import primepi, integer_nthroot, primerange
def A030513(n):
    def f(x): return int(n+x-primepi(integer_nthroot(x,3)[0])+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 16 2024

{n : A000005(n) = 4}. - Juri-Stepan Gerasimov, Oct 10 2009

Incorrect comments removed by Charles R Greathouse IV, Mar 18 2010

A003680 Smallest number with 2n divisors.

Original entry on oeis.org

2, 6, 12, 24, 48, 60, 192, 120, 180, 240, 3072, 360, 12288, 960, 720, 840, 196608, 1260, 786432, 1680, 2880, 15360, 12582912, 2520, 6480, 61440, 6300, 6720, 805306368, 5040, 3221225472, 7560, 46080, 983040, 25920, 10080, 206158430208, 3932160, 184320, 15120

Offset: 1

N. J. A. Sloane, Mira Bernstein

Refers to the least number which is multiplicatively n-perfect, i.e. least number m the product of whose divisors equals m^n. - Lekraj Beedassy, Sep 18 2004

For n=1 to 5, a(n) equals second term of A008578, A007422, A162947, A048945, A030628. - Michel Marcus, Feb 04 2014

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 23.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..3300 (terms 1..1000 from T. D. Noe using A005179)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A005179 (n), A061283 (2n-1), A118224 (at least 2n).

A005179 = Cases[Import["https://oeis.org/A005179/b005179.txt", "Table"], {, }][[All, 2]];
A = {#, DivisorSigma[0, #]}& /@ A005179;
a[n_] := SelectFirst[A, #[[2]] == 2n&][[1]];
a /@ Range[1000] (* Jean-François Alcover, Nov 10 2019 *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[m < n, {}, {{n}}]; mp[n_, m_] := Join @@ Table[Map[Prepend[#, d] &, mp[n/d, d]], {d, Select[Rest[Divisors[n]], # <= m &]}]; mp[n_] := mp[n, n]; Table[mulpar = mp[2*n] - 1; Min[Table[Product[Prime[s]^mulpar[[j, s]], {s, 1, Length[mulpar[[j]]]}], {j, 1, Length[mulpar]}]], {n, 1, 100}] (* Vaclav Kotesovec, Apr 04 2021 *)
With[{tbl=Table[{n,DivisorSigma[0,n]},{n,800000}]},Table[SelectFirst[tbl,#[[2]]==2k&],{k,20}]][[;;,1]] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Jul 06 2025 *)

a(n)=my(k=2*n); while(numdiv(k)!=2*n, k++); k \\ Charles R Greathouse IV, Jun 23 2017

from sympy import divisors
def a(n):
  m = 4*n - 2
  while len(divisors(m)) != 2*n: m += 1
  return m
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Feb 06 2021

Bisection of A005179(n). - Lekraj Beedassy, Sep 21 2004

More terms from Jud McCranie Oct 15 1997

A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are distinct primes.

Original entry on oeis.org

1, 48, 80, 112, 162, 176, 208, 272, 304, 368, 405, 464, 496, 512, 567, 592, 656, 688, 752, 848, 891, 944, 976, 1053, 1072, 1136, 1168, 1250, 1264, 1328, 1377, 1424, 1539, 1552, 1616, 1648, 1712, 1744, 1808, 1863, 1875, 2032, 2096, 2192, 2224, 2349, 2384

Offset: 1

Jeff Burch

Also 1 together with numbers with 10 divisors. Also numbers n such that product of all proper divisors of n equals n^4.

If M(n) denotes the product of all divisors of n, then n is said to be k-multiplicatively perfect if M(n)=n^k. All such numbers are of the form p*q^(k-1) or p^(2k-1). The sequence A030628 is therefore 5-multiplicatively perfect. See the Links for A007422. - Walter Kehowski, Sep 13 2005

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston, MA, 1976. p. 119.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, entry for 48, page 106, 1997.

R. J. Mathar, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Divisor Product
Index to sequences related to prime signature

Cf. A030515, A030627, A030629.

with(numtheory): k:=5: MPL:=[]: for z from 1 to 1 do for n from 1 to 5000 do if convert(divisors(n),`*`) = n^k then MPL:=[op(MPL),n] fi od; od; MPL; # Walter Kehowski, Sep 13 2005

Join[{1},Select[Range[6000],DivisorSigma[0,#]==10&]] (* Vladimir Joseph Stephan Orlovsky, May 05 2011 *)
Select[Range[2500],Times@@Most[Divisors[#]]==#^4&] (* Harvey P. Dale, Nov 04 2024 *)

{v=[]; for(n=1,500,v=concat(v, if(numdiv(n)==10,n,",")); ); v} \\ Jason Earls, Jun 18 2001

list(lim)=my(v=List([1]), t); forprime(p=2, (lim\2+.5)^(1/4), t=p^4; forprime(q=2, lim\t, if(p==q, next); listput(v, t*q))); forprime(p=2,(lim+.5)^(1/9),listput(v,p^9)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Apr 26 2012

from sympy import primepi, primerange, integer_nthroot
def A030628(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n-1+x-sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))+primepi(integer_nthroot(x,5)[0])-primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

Union A178739 U A179665 {1}. - R. J. Mathar, Apr 03 2011

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Dec 23 2001

More terms from Walter Kehowski, Sep 13 2005

A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

Original entry on oeis.org

1, 2, 3, 8, 5, 1, 7, 1, 27, 1, 11, 48, 13, 1, 1, 16, 17, 162, 19, 80, 1, 1, 23, 16, 125, 1, 1, 112, 29, 25, 31, 512, 1, 1, 1, 1944, 37, 1, 1, 25, 41, 49, 43, 176, 405, 1, 47, 48, 343, 1250, 1, 208, 53, 324, 1, 49, 1, 1, 59, 9, 61, 1, 567, 8, 1, 121, 67, 272, 1, 49, 71, 9, 73, 1

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

a(n) = r(n,tau(n)), where r is defined as follows:
let d(n,j) = j-th divisor of n, 1 <= j <= tau(n) = A000005(n), r(n,1)=d(n,1), r(n,j) = if d(n,j) divides r(n,j-1) then r(n,j-1)/d(n,j) else r(n,j-1)*d(n,j), 1 < j <= tau(n);
p prime: a(p)=p, a(p^2)=p^3, a(p^3)=1, a(p^k)=p^A008344(k+1);
a(m)=1 iff m multiplicatively perfect: a(A007422(k))=1.
a(A084111(n)) = A084111(n). - Reinhard Zumkeller, Jul 31 2014

			Divisors of 48 = {1,2,3,4,6,8,12,16,24,48}: 1*2*3 = 6 -> 6*4 = 24 -> 24/6 = 4 -> 4*8 = 32 -> 32*12 = 384 -> 384/16 = 24 -> 24/24 = 1 -> 1*48 = a(48);
divisors of 49 = {1,7,49}: 1*7 = 7 -> 7*49 = 343 = a(49);
divisors of 50 = {1,2,5,10,25,50}: 1*2*5 = 10 -> 10/10 = 1 -> 1*25 = 25 -> 25*50 = 1250 = a(50).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Divisor Product.
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A027750, A084111 (fixed points), A084113, A084114.

a084110 = foldl (/*) 1 . a027750_row where
   x /* y = if m == 0 then x' else x*y where (x',m) = divMod x y
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

a[n_] := Module[{d = Divisors[n], e}, e[i_] := e[i] = If[i == 1, 1, If[Divisible[e[i-1], d[[i]]], e[i-1]/d[[i]], e[i-1] d[[i]]]]; e[Length[d]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 10 2021 *)

Corrected and extended by David Wasserman, Dec 14 2004

A058080 Numbers whose product of divisors exceeds their square.

Original entry on oeis.org

12, 16, 18, 20, 24, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 126, 128, 130, 132, 135, 136, 138, 140, 144, 147

Offset: 1

N. J. A. Sloane, Nov 24 2000

nonn

Numbers with five or more divisors. - Lekraj Beedassy, Sep 11 2003

Called multiplicatively abundant numbers by Chau (2004). - Amiram Eldar, Jun 29 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
William Chau, The tau, sigma, rho functions, and some related numbers, Pi Mu Epsilon Journal, Vol. 11, No. 10 (Spring 2004), pp. 519-534; entire issue.

Complement of A007964.

Cf. A000430, A007422, A007955.

Select[Range[150], #^(DivisorSigma[0, #]/2) > #^2 &] (* Amiram Eldar, Jun 29 2022 *)
Select[Range[200],Times@@Divisors[#]>#^2&] (* Harvey P. Dale, Oct 20 2024 *)

is(n)=numdiv(n)>4 \\ Charles R Greathouse IV, Sep 18 2015

from sympy import divisor_count
def ok(n): return divisor_count(n) > 4
print([k for k in range(148) if ok(k)]) # Michael S. Branicky, Dec 16 2021

The number of terms not exceeding x is N(x) ~ x*(1 - log(log(x))/log(x)) (Chau, 2004). - Amiram Eldar, Jun 29 2022

A084116 Numbers m such that A084115(m) = 1.

Original entry on oeis.org

2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 27, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119

Offset: 1

Reinhard Zumkeller, May 12 2003

nonn

A084113(a(n)) = A084114(a(n)) + 1.
Union of primes and multiplicatively perfect numbers (A000040, A007422).
A084115(a(n)) = 1; A066729(a(n)) = a(n).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Multiplicative Perfect Number.

Cf. A084110, A066729, A084113, A084114, A084115, A066423 (complement).

a084116 n = a084116_list !! (n-1)
a084116_list = filter ((== 1) . a084115) [1..]
-- Reinhard Zumkeller, Jul 31 2014

Select[Range[2, 200], PrimeQ[DivisorSigma[0, #]^DivisorSigma[0, #] + 1] &] (* Carl Najafi, Oct 19 2011 *)

is(n)=isprime(n) || numdiv(n) == 4 \\ Charles R Greathouse IV, Oct 19 2015

It appears that a(n) = n such that A000005(n)^A000005(n)+1 is prime. - Carl Najafi, Oct 19 2011

Corrected and edited by Carl Najafi, Oct 19 2011

Revised by Reinhard Zumkeller, Jul 31 2014

A036455 Numbers n such that d(d(n)) is an odd prime, where d(k) is the number of divisors of k.

Original entry on oeis.org

6, 8, 10, 14, 15, 21, 22, 26, 27, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 168, 177, 178, 183

Offset: 1

Labos Elemer

nonn

Compare with sequence A007422 and A030513 -- the resemblance is rather strong. Still this sequence is different. For example, 36, 100, 120, and 168 are here.

			a(15) = 39 and d(39) = 4, d(d(39)) = d(4) = 3 and d(d(d(39))) = 2. After 3 iteration the equilibrium is reached.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A007422, A030513, A036450, A036452, A036454, A036456, A036457, A036458.

filter:= proc(n) local r;
  r:= numtheory:-tau(numtheory:-tau(n));
  r::odd and isprime(r)
end proc:
select(filter, [$1..1000]); # Robert Israel, Feb 02 2016

fQ[n_] := Module[{d2 = DivisorSigma[0, DivisorSigma[0, n]]}, d2 > 2 && PrimeQ[d2]]; Select[Range[200], fQ] (* T. D. Noe, Jan 22 2013 *)

is(n)=isprime(n=numdiv(numdiv(n))) && n>2 \\ Charles R Greathouse IV, Jan 22 2013

d(d(d(a(n)))) = 2 for all n.

A036459(a(n)) = 3. - Ivan Neretin, Jan 25 2016

Definition clarified by R. J. Mathar and Charles R Greathouse IV, Jan 22 2013

A048741 Product of aliquot divisors of composite n (1 and primes omitted).

Original entry on oeis.org

2, 6, 8, 3, 10, 144, 14, 15, 64, 324, 400, 21, 22, 13824, 5, 26, 27, 784, 27000, 1024, 33, 34, 35, 279936, 38, 39, 64000, 74088, 1936, 2025, 46, 5308416, 7, 2500, 51, 2704, 157464, 55, 175616, 57, 58, 777600000, 62, 3969, 32768, 65, 287496, 4624, 69

Offset: 1

Enoch Haga

			The third composite number is 8, for which the product of aliquot divisors is 4*2*1 = 8, so a(3)=8.

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed., pages 10, 23. New York: Dover, 1966. ISBN 0-486-21096-0.

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

This is A007956 omitting the 1's.

Cf. A002808, A007422, A007956, A048740.

Composite[n_] := FixedPoint[n + PrimePi[ # ] + 1 &, n + PrimePi[n] + 1]; Table[ Times @@ Select[ Divisors[ Composite[n]], # < Composite[n] & ], {n, 1, 60} ]
pd[n_] := n^(DivisorSigma[0, n]/2 - 1); pd /@ Select[Range[100], CompositeQ] (* Amiram Eldar, Sep 07 2019 *)

a(n) = A007956(A002808(n)). - Michel Marcus, Sep 07 2019

a(33) inserted by Amiram Eldar, Sep 07 2019

A162947 Numbers k such that the product of all divisors of k equals k^3.

Original entry on oeis.org

1, 12, 18, 20, 28, 32, 44, 45, 50, 52, 63, 68, 75, 76, 92, 98, 99, 116, 117, 124, 147, 148, 153, 164, 171, 172, 175, 188, 207, 212, 236, 242, 243, 244, 245, 261, 268, 275, 279, 284, 292, 316, 325, 332, 333, 338, 356, 363, 369, 387, 388, 404, 412, 423, 425, 428

Offset: 1

Claudio Meller, Jul 18 2009

Contains the terms of A054753 (products p*q^2 of a prime p and the square of a different prime q), 1, and p^5, where p is prime.

Numbers k such that k^2 is equal to the product of proper divisors of k. - Juri-Stepan Gerasimov, May 03 2011

			18 is in the sequence because the product of its divisors is 1 * 2 * 3 * 6 * 9 * 18 = 18^3.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
David A. Corneth, PARI program

Cf. A111398, A030628. - R. J. Mathar, Jul 19 2009

Cf. A008578 (product of divisors equals n), A007422 (product of divisors equals n^2).

Select[Range[500], Surd[Times@@Divisors[#], 3] == # &] (* Harvey P. Dale, Mar 15 2017 *)
Join[{1},Select[Range[430],DivisorSigma[0,#]==6 &]] (* Stefano Spezia, Jul 14 2025 *)

isok(n) = my(d = divisors(n)); prod(i=1, #d, d[i]) == n^3; \\ Michel Marcus, Feb 04 2014

PARI
```
\\ See Corneth link
```

from itertools import chain, count, islice
from sympy import divisor_count
def A162947_gen(): # generator of terms
    return chain((1,),filter(lambda n:divisor_count(n)==6,count(2)))
A162947_list = list(islice(A162947_gen(),20)) # Chai Wah Wu, Jun 25 2022

{n: A007955(n) = A000578(n)}. - R. J. Mathar, Jul 19 2009

{1} UNION A030515. - R. J. Mathar, Jul 19 2009

Edited by R. J. Mathar, Jul 19 2009

cp's OEIS Frontend

A007956 Product of the proper divisors of n.

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A030513 Numbers with 4 divisors.

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A003680 Smallest number with 2n divisors.

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A030628 1 together with numbers of the form p*q^4 and p^9, where p and q are distinct primes.

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A084110 Let L(n) = ordered list of divisors of n = {d_1=1, d_2, ..., d_k=n}; set e_1=1, e_i = e_{i-1}/d_i if that is an integer otherwise e_i = e_{i-1}*d_i; then a(n) = e_k.

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A058080 Numbers whose product of divisors exceeds their square.

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