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A005179
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Smallest number with exactly n divisors.
(Formerly M1026)
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110
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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
(list;
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OFFSET
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1,2
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COMMENTS
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Also the smallest positive number with n-1 proper divisors. - Roderick MacPhee, Dec 11 2012
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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.
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, 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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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].
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.
Eric Weisstein's World of Mathematics, Divisor
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FORMULA
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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, Jul 15 2004
a(p^n)=(2*3...*p_n)^(p-1) for p > log p_n / log 2. Unpublished proof from Andrzej Schinzel. - Tomasz 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). - Tomasz Ordowski, Aug 30 2005
If n=2^k then a(2^k)=A037992(k). - Tomasz 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
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PROG
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(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, May 26 2008 */
(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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CROSSREFS
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Cf. A007416, A099316, A003586, A099311, A099313, A050376 and A037992.
Cf. A061799.
Sequence in context: A209867 A136033 A099315 * A037019 A096174 A096173
Adjacent sequences: A005176 A005177 A005178 * A005180 A005181 A005182
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane, David Singmaster
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EXTENSIONS
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More terms from David W. Wilson
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STATUS
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approved
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