A262981 -id:A262981 - cp's OEIS frontend

A005179 Smallest number with exactly n divisors.

1, 2, 4, 6, 16, 12, 64, 24, 36, 48, 1024, 60, 4096, 192, 144, 120, 65536, 180, 262144, 240, 576, 3072, 4194304, 360, 1296, 12288, 900, 960, 268435456, 720, 1073741824, 840, 9216, 196608, 5184, 1260, 68719476736, 786432, 36864, 1680, 1099511627776, 2880

Offset: 1

N. J. A. Sloane, David Singmaster

A number n is called ordinary iff a(n)=A037019(n). Brown shows that the ordinary numbers have density 1 and all squarefree numbers are ordinary. See A072066 for the extraordinary or exceptional numbers. - M. F. Hasler, Oct 14 2014

All terms are in A025487. Therefore, a(n) is even for n > 1. - David A. Corneth, Jun 23 2017 [corrected by Charles R Greathouse IV, Jul 05 2023]

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.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 52.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 89.

Matthew House, Table of n, a(n) for n = 1..3322 (terms 1..2000 from Don Reble)
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].
Ron Brown, The minimal number with a given number of divisors, Journal of Number Theory 116 (2006) 150-158.
M. E. Grost, The smallest number with a given number of divisors, Amer. Math. Monthly, 75 (1968), 725-729.
Joe Roberts, Lure of the Integers, Annotated scanned copy of pp. 81, 86 with notes.
Anna K. Savvopoulou and Christopher M. Wedrychowicz, On the smallest number with a given number of divisors, The Ramanujan Journal, 2015, Vol. 37, pp. 51-64.
David Singmaster, Letter to N. J. A. Sloane, Oct 03 1982.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Divisor.
Robert G. Wilson v, Letter to N. J. A. Sloane, Dec 17 1991.

Cf. A000005, A007416, A099316, A003586, A025487, A099311, A099313, A050376, A037992, A061799, A162247, A262981, A262983.

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a005179 n = succ $ fromJust $ elemIndex n $ map a000005 [1..]
-- Reinhard Zumkeller, Apr 01 2011

A005179_list := proc(SearchLimit, ListLength)
local L, m, i, d; m := 1;
L := array(1..ListLength,[seq(0,i=1..ListLength)]);
for i from 1 to SearchLimit while m <= ListLength do
  d := numtheory[tau](i);
  if d <= ListLength and 0 = L[d] then L[d] := i;
  m := m + 1; fi
od:
print(L) end: A005179_list(65537,18);
# If a '0' appears in the list the search limit has to be increased. - Peter Luschny, Mar 09 2011
# alternative
# Construct list of ordered lists of factorizations of n with
# minimum divisors mind.
# Returns a list with A001055(n) entries if called with mind=2.
# Example: print(ofact(10^3,2))
ofact := proc(n,mind)
    local fcts,d,rec,r ;
    fcts := [] ;
    for d in numtheory[divisors](n) do
        if d >= mind then
            if d = n then
                fcts := [op(fcts),[n]] ;
            else
                # recursive call supposed one more factor fixed now
                rec := procname(n/d,max(d,mind)) ;
                for r in rec do
                    fcts := [op(fcts),[d,op(r)]] ;
                end do:
            end if;
        end if;
    end do:
    return fcts ;
end proc:
A005179 := proc(n)
    local Lexp,a,eList,cand,maxxrt ;
    if n = 1 then
        return 1;
    end if;
    Lexp := ofact(n,2) ;
    a := 0 ;
    for eList in Lexp do
        maxxrt := ListTools[Reverse](eList) ;
        cand := mul( ithprime(i)^ ( op(i,maxxrt)-1),i=1..nops(maxxrt)) ;
        if a =0 or cand < a then
            a := cand ;
        end if;
    end do:
    a ;
end proc:
seq(A005179(n),n=1..40) ; # R. J. Mathar, Jun 06 2024

a = Table[ 0, {43} ]; Do[ d = Length[ Divisors[ n ]]; If[ d < 44 && a[[ d ]] == 0, a[[ d]] = n], {n, 1, 1099511627776} ]; a
(* Second program: *)
Function[s, Map[Lookup[s, #] &, Range[First@ Complement[Range@ Max@ #, #] - 1]] &@ Keys@ s]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)
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[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 *)
a[n_] := Module[{e = f[n] - 1}, Min[Times @@@ ((Prime[Range[Length[#], 1, -1]]^#) & /@ e)]]; Array[a, 100] (* Amiram Eldar, Jul 26 2025 using the function f by T. D. Noe at A162247 *)

(prodR(n,maxf)=my(dfs=divisors(n),a=[],r); for(i=2,#dfs, if( dfs[i]<=maxf, if(dfs[i]==n, a=concat(a,[[n]]), r=prodR(n/dfs[i],min(dfs[i],maxf)); for(j=1,#r, a=concat(a,[concat(dfs[i],r[j])]))))); a); A005179(n)=my(pf=prodR(n,n),a=1,b); for(i=1,#pf, b=prod(j=1,length(pf[i]),prime(j)^(pf[i][j]-1)); if(bA005179(n)", ")) \\ R. J. Mathar, May 26 2008, edited by M. F. Hasler, Oct 11 2014

from math import prod
from sympy import isprime, divisors, prime
def A005179(n):
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

a(p) = 2^(p-1) for primes p: a(A000040(n)) = A061286(n); a(p^2) = 6^(p-1) for primes p: a(A001248(n)) = A061234(n); a(p*q) = 2^(q-1)*3^(p-1) for primes p<=q: a(A001358(n)) = A096932(n); a(p*m*q) = 2^(q-1) * 3^(m-1) * 5^(p-1) for primes pA005179(A007304(n)) = A061299(n). - Reinhard Zumkeller, Jul 15 2004 [This can be continued to arbitrarily many distinct prime factors since no numbers in A072066 (called "exceptional" or "extraordinary") are squarefree. - Jianing Song, Jul 18 2025]
a(p^n) = (2*3...*p_n)^(p-1) for p > log p_n / log 2. Unpublished proof from Andrzej Schinzel. - Thomas Ordowski, Jul 22 2005
If p is a prime and n=p^k then a(p^k)=(2*3*...*s_k)^(p-1) where (s_k) is the numbers of the form q^(p^j) for every q and j>=0, according to Grost (1968), Theorem 4. For example, if p=2 then a(2^k) is the product of the first k members of the A050376 sequence: number of the form q^(2^j) for j>=0, according to Ramanujan (1915). - Thomas Ordowski, Aug 30 2005
a(2^k) = A037992(k). - Thomas Ordowski, Aug 30 2005
a(n) <= A037019(n) with equality except for n in A072066. - M. F. Hasler, Jun 15 2022

More terms from David W. Wilson

A262983 Terms of A005179 divisible by their indices in order of appearance in A005179.

Original entry on oeis.org

1, 2, 12, 24, 36, 60, 180, 240, 360, 720, 1260, 1680, 3600, 6720, 5040, 10080, 32400, 15120, 20160, 25200, 60480, 55440, 810000, 100800, 181440, 110880, 226800, 221760, 277200, 907200, 665280, 1587600, 720720, 5670000, 1108800, 3548160, 1995840, 1441440, 2494800, 6350400

Offset: 1

Vladimir Letsko, Oct 06 2015

A005179(n) is in this sequence iff it is divisible by n. Thus this is a subsequence of A005179 indexed by A262981.
Also this sequence is the intersection of A033950 and A005179. Hence this sequence has density zero. - Vladimir Letsko, Dec 16 2016
It seems that this sequence is a subsequence of A262981.
This sequence is not in ascending order as terms of A005179 divisible by their number of divisors do not occur in ascending order. For terms sorted in ascending order see A110821. - David A. Corneth, Dec 10 2021

			12 is a term since it is the smallest positive integer having exactly 6 divisors and divisible by 6.

David A. Corneth, Table of n, a(n) for n = 1..10000
Vladimir Letsko, The table of correspondence between A262983 and A262981

Cf. A000005, A033950, A005179, A110821, A262981, A279373.

Take[#, 33] &@ DeleteCases[#, 0] &@ Function[s, ReplacePart[#, Flatten@ Map[{# -> Function[k, k Boole[Divisible[k, #]]]@ Lookup[s, #]} &, Keys@ s]] &@ ConstantArray[0, Max@ Keys@ s]]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)

a(n) = A005179(A262981(n)).

A000005(a(n)) = A262981(n).

Name clarified by David A. Corneth, Dec 10 2021

A279373 Numbers n such that number of divisors of n divides n and at the same time the least number having exactly n divisors is divisible by n.

Original entry on oeis.org

1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 180, 225, 240, 252, 288, 360, 396, 441, 448, 450, 504, 560, 600, 625, 672, 720, 792, 880, 882, 936, 1040, 1056, 1200, 1248, 1250, 1260, 1344, 1408, 1440, 1620, 1664, 1680, 1800, 1980, 2000, 2016, 2025, 2160, 2176, 2240, 2340, 2640, 2700, 2772, 3120, 3168

Offset: 1

Vladimir Letsko, Dec 11 2016

nonn

Intersection of A033950 and A262981.

			8 is in the sequence because 8 is divisible by tau(8) and at the same time 8 divides 24 which is the least number having exactly 8 divisors.

Amiram Eldar, Table of n, a(n) for n = 1..267
A. Bundy, Simon Colton, T. Walsh, HR - A system for Machine Discovery in Finite Algebras, ECAI 1998.
S. Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2, 1999, #2.
S. 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, 13:2 (1990), pp. 383-386.
Claudia Spiro, How often is the number of divisors of n a divisor of n?, J. Number Theory 21 (1985), no. 1, 81-100.
Vladimir Letsko, Mathematical Marathon, Problem 216 (in Russian)

Cf. A000005, A005179, A033950, A262981, A262983.

Function[s, Select[TakeWhile[#, KeyExistsQ[s, #] &], Divisible[Lookup[s, #], #] &] &@ Select[Range@ 3000, Divisible[#, DivisorSigma[0, #]] &]]@ Map[First, KeySort@ PositionIndex@ Table[DivisorSigma[0, n], {n, 10^7}]] (* Michael De Vlieger, Dec 11 2016, Version 10 *)

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A005179 Smallest number with exactly n divisors.

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A262983 Terms of A005179 divisible by their indices in order of appearance in A005179.

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A279373 Numbers n such that number of divisors of n divides n and at the same time the least number having exactly n divisors is divisible by n.

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Mathematica