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
References
- 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.
Links
- 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.
Crossrefs
Programs
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Haskell
import Data.List (elemIndex) import Data.Maybe (fromJust) a005179 n = succ $ fromJust $ elemIndex n $ map a000005 [1..] -- Reinhard Zumkeller, Apr 01 2011
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Maple
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
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Mathematica
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 *) -
PARI
(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
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Python
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 1
Formula
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
Extensions
More terms from David W. Wilson
Comments