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A005179 Smallest number with exactly n divisors.
(Formerly M1026)
110
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

Also the smallest positive number with n-1 proper divisors. - Roderick MacPhee, Dec 11 2012

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

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.

T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Eric Weisstein's World of Mathematics, Divisor

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<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 Unpublished proof from Andrzej Schinzel.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

a(2^k) = A037992(k). - Tomasz Ordowski, Aug 30 2005

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

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<a || i==1, a=b)); a} /* for(n=1, 100, print1(A005179(n)", ")) */ \\ R. J. Mathar, May 26 2008, edited by M. F. Hasler, Oct 11 2014

(Haskell)

import Data.List (elemIndex)

import Data.Maybe (fromJust)

a005179 n = succ $ fromJust $ elemIndex n $ map a000005 [1..]

-- Reinhard Zumkeller, Apr 01 2011

CROSSREFS

Cf. A007416, A099316, A003586, A099311, A099313, A050376, A037992, A061799.

Sequence in context: A209867 A136033 A099315 * A037019 A096174 A096173

Adjacent sequences:  A005176 A005177 A005178 * A005180 A005181 A005182

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, David Singmaster

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 21 18:50 EST 2014. Contains 252325 sequences.