

A038547


Least number with exactly n odd divisors.


40



1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 315, 531441, 3645, 2025, 945, 43046721, 1575, 387420489, 2835, 18225, 295245, 31381059609, 3465, 50625, 2657205, 11025, 25515, 22876792454961, 14175, 205891132094649, 10395, 1476225, 215233605
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OFFSET

1,2


COMMENTS

Also least odd number with exactly n divisors.  Lekraj Beedassy, Aug 30 2006
a(2n1) = {1,9,81,729,225,59049,...} are the squares. A122842(n) = sqrt(a(2n1)) = {1,3,9,27,15,243,729,45,6561,19683,135,177147,225,105,4782969,14348907,1215,...}.  Alexander Adamchuk, Sep 13 2006
Also the least number k such that there are n partitions of k whose elements are consecutive integers. I.e., 1=1, 3=1+2=3, 9=2+3+4=4+5=9, 15=1+2+3+4+5=4+5+6=7+8=15, etc.  Robert G. Wilson v, Jun 02 2007
The politeness of an integer, A069283(n), is defined to be the number of its nontrivial runsum representations, and the sequence 3, 9, 15, 81, 45, 729, 105, ... represents the least integers to have a politeness of 1, 2, 3, 4, ... This is also the sequence of smallest integers with n+1 odd divisors and so apart from the leading 1, is precisely this sequence.  Ant King, Sep 23 2009
a(n) is also the least number k with the property that the symmetric representation of sigma(k) has n subparts.  Omar E. Pol, Dec 31 2016


LINKS

Don Reble, Table of n, a(n) for n = 1..2000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.


FORMULA

a(p) = 3^(p1) for primes p.  Zak Seidov, Apr 18 2006
a(n) = A119265(n,n).  Reinhard Zumkeller, May 11 2006
It was suggested by Alexander Adamchuk that for all n >= 1, we have a(3^(n1)) = (p(n)#/2)^2 = (A002110(n)/2)^2 = A070826(n)^2. But this is false! E.g., (p(n)#/2)^2=3^2 5^2 7^2 ...23^2 29^2 does indeed have 3^9 odd factors, but it is greater than 3^8*5^2 7^2 ...23^2 which has 9*3*3*3*3*3*3*3 = 9*3^7=3^9 odd factors.  Richard Sabey, Oct 06 2007.
a(A053640(m)) = a(A000005(A053624(m))) = A053624(m).  Rick L. Shepherd, Apr 20 2008


MATHEMATICA

Table[Select[Range[1, 532000, 2], DivisorSigma[0, #]==k+1 &, 1], {k, 0, 15}]//Flatten (* Ant King, Nov 28 2010 *)
2#1&/@With[{ds=DivisorSigma[0, Range[1, 600000, 2]]}, Table[Position[ds, n, 1, 1], {n, 16}]]//Flatten (* The program is not suitable for generating terms beyond a(16) *) (* Harvey P. Dale, Jun 06 2017 *)


PROG

(PARI) for(nd=1, 15, forstep(k=1, 10^66, 2, if(nd==numdiv(k), print1(k, ", "); break())))
(Haskell)
import Data.List (find)
import Data.Maybe (fromJust)
a038547 n = fromJust $ find ((== n) . length . divisors) [1, 3..]
where divisors m = filter ((== 0) . mod m) [1..m]
 Reinhard Zumkeller, Feb 24 2011


CROSSREFS

A122842 = Sqrt[ a(2n1) ].
Cf. A001227, A005179, A002110, A070826, A000005, A053640, A053624, A237593, A279387.
Row 1 of A266531.  Omar E. Pol, Dec 31 2016
Sequence in context: A192165 A050869 A323679 * A348199 A242438 A355716
Adjacent sequences: A038544 A038545 A038546 * A038548 A038549 A038550


KEYWORD

nonn,nice,changed


AUTHOR

Tom Verhoeff


EXTENSIONS

Corrected by Ron Knott, Feb 22 2001
a(30) from Zak Seidov, Apr 18 2006
a(32)a(34) from Lekraj Beedassy, Aug 30 2006


STATUS

approved



