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 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Also least odd number with exactly n divisors. - Lekraj Beedassy, Aug 30 2006 a(2n-1) = {1,9,81,729,225,59049,...} are the squares. A122842(n) = sqrt(a(2n-1)) = {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^(p-1) 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^(n-1)) = (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(2n-1) ]. 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 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

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Last modified October 6 21:32 EDT 2022. Contains 357270 sequences. (Running on oeis4.)