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A005179
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Smallest number with exactly n divisors.
(Formerly M1026)
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215
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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
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OFFSET
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1,2
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COMMENTS
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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
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REFERENCES
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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.
J. 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).
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LINKS
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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].
Eric Weisstein's World of Mathematics, Divisor
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FORMULA
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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 Unpublished proof from Andrzej Schinzel.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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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(b<a || i==1, a=b)); a
(Haskell)
import Data.List (elemIndex)
import Data.Maybe (fromJust)
a005179 n = succ $ fromJust $ elemIndex n $ map a000005 [1..]
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CROSSREFS
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Cf. A000005, A007416, A099316, A003586, A025487, A099311, A099313, A050376, A037992, A061799, A262981, A262983.
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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