A108687 Numbers of the form (9^i)*(11^j), with i, j >= 0.

1, 9, 11, 81, 99, 121, 729, 891, 1089, 1331, 6561, 8019, 9801, 11979, 14641, 59049, 72171, 88209, 107811, 131769, 161051, 531441, 649539, 793881, 970299, 1185921, 1449459, 1771561, 4782969, 5845851, 7144929, 8732691, 10673289, 13045131

Offset: 1

Author

Douglas Winston (douglas.winston(AT)srupc.com), Jun 17 2005

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A003597.

Cf. A025633, A025634, A107788, A107764, A003596, A107988, A003598, A003599, A108090.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a108687 n = a108687_list !! (n-1)
a108687_list = f $ singleton (1,0,0) where
   f s = y : f (insert (9 * y, i + 1, j) $ insert (11 * y, i, j + 1) s')
         where ((y, i, j), s') = deleteFindMin s
-- Reinhard Zumkeller, May 15 2015

Mathematica

f[upto_]:=With[{max9=Floor[Log[9,upto]],max11=Floor[Log[11,upto]]}, Select[Union[Times@@{9^First[#],11^Last[#]}&/@Tuples[{Range[0, max9], Range[0, max11]}]], #<=upto&]]; f[14000000]  (* Harvey P. Dale, Mar 11 2011 *)

Python

from sympy import integer_log
def A108687(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(integer_log(x//11**i,9)[0]+1 for i in range(integer_log(x,11)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Mar 25 2025

Formula

Sum_{n>=1} 1/a(n) = (9*11)/((9-1)*(11-1)) = 99/80. - Amiram Eldar, Sep 24 2020

a(n) ~ exp(sqrt(2*log(9)*log(11)*n)) / sqrt(99). - Vaclav Kotesovec, Sep 24 2020