Search: id:A005179
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%I A005179 M1026
%S A005179 1,2,4,6,16,12,64,24,36,48,1024,60,4096,192,144,120,65536,180,262144,240,
%T A005179 576,3072,4194304,360,1296,12288,900,960,268435456,720,1073741824,840,
%U A005179 9216,196608,5184,1260,68719476736,786432,36864,1680,1099511627776,2880
%N A005179 Smallest number with exactly n divisors.
%D A005179 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.
%D A005179 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.
%D A005179 M. E. Grost, The smallest number with a given number of divisors, Amer. 
               Math. Monthly, 75 (1968), 725-729.
%D A005179 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 86.
%D A005179 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005179 Don Reble, <a href="b005179.txt">Table of n, a(n) for n = 1..2000</a>
%H A005179 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A005179 T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Rectangular and Trapezoidal Arrangements</a>, J. Integer Sequences, 
               Vol. 2, 1999, #99.1.6.
%H A005179 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Divisor.html">Link to a section of The World of Mathematics.</a>
%F A005179 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<m<q: A005179(A007304(n)) = A061299(n). 
               - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 15 2004
%F A005179 a(p^n)=(2*3...*p_n)^(p-1) for p > log p_n / log 2. Unpublished proof 
               from Andrzej Schinzel. - Tomasz Ordowski (ordot(AT)poczta.onet.pl), 
               Jul 22 2005
%F A005179 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). - Tomasz 
               Ordowski (ordot(AT)poczta.onet.pl), Aug 30 2005
%F A005179 If n=2^k then a(2^k)=A037992(k). - Tomasz Ordowski (ordot(AT)poczta.onet.pl), 
               Aug 30 2005
%t A005179 a = Table[ 0, {43} ]; Do[ d = Length[ Divisors[ n ]]; If[ d < 44 && a[[ 
               d ]] == 0, a[[ d]] = n], {n, 1, 1099511627776} ]; a
%o A005179 (PARI) prodR(n,maxf)={ local(dfs,a=[],r,tmp ) ; dfs=divisors(n) ; for(i=2,
               length(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,length(r), tmp=concat(dfs[i],r[j]) 
               ; a=concat(a,[tmp]) ; ) ; ) ; ) ; ) ; return(a) ; } A005179(n)={ 
               local(pf=prodR(n,n),a=1,b) ; for(i=1,length(pf), b=prod(j=1,length(pf[i]),
               prime(j)^(pf[i][j]-1)) ; if(b<a || i==1, a=b ) ; ) ; return(a) ; 
               } { for(n=1,100, print1(A005179(n)",") ; ) } /* R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               May 26 2008 */
%Y A005179 Cf. A007416, A099316, A003586, A099311, A099313.
%Y A005179 Cf. A050376 and A037992.
%Y A005179 Sequence in context: A049022 A136033 A099315 this_sequence A037019 A096174 
               A096173
%Y A005179 Adjacent sequences: A005176 A005177 A005178 this_sequence A005180 A005181 
               A005182
%K A005179 nonn,nice,easy
%O A005179 1,2
%A A005179 N. J. A. Sloane (njas(AT)research.att.com), David Singmaster
%E A005179 More terms from David W. Wilson (davidwwilson(AT)comcast.net)

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