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 A005179 Smallest number with exactly n divisors. (Formerly M1026) 215
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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] 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. 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). LINKS Don Reble, Table of n, a(n) for n = 1..2000 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]. R. 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. J. 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, #99.1.6. Eric Weisstein's World of Mathematics, Divisor R. G. Wilson v, Letter to N. J. A. Sloane, Dec 17 1991. 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 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 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 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 *) PROG (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

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Last modified May 19 14:45 EDT 2024. Contains 372698 sequences. (Running on oeis4.)