A003594 Numbers of the form 3^i*7^j with i, j >= 0.

1, 3, 7, 9, 21, 27, 49, 63, 81, 147, 189, 243, 343, 441, 567, 729, 1029, 1323, 1701, 2187, 2401, 3087, 3969, 5103, 6561, 7203, 9261, 11907, 15309, 16807, 19683, 21609, 27783, 35721, 45927, 50421, 59049, 64827, 83349, 107163, 117649

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 70 terms from Vincenzo Librandi)
• Vaclav Kotesovec, Graph - the asymptotic ratio (600000 terms).

Crossrefs

Cf. A003586, A003591, A003592, A003593, A003595.

Cf. A025642, A025665.

Programs

GAP

Filtered([1..120000],n->PowerMod(21,n,n)=0); # Muniru A Asiru, Mar 19 2019

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a003594 n = a003594_list !! (n-1)
a003594_list = f $ singleton 1 where
   f s = y : f (insert (3 * y) $ insert (7 * y) s')
               where (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 16 2015

Magma

[n: n in [1..120000] | PrimeDivisors(n) subset [3,7]]; // Bruno Berselli, Sep 24 2012

Mathematica

f[upto_]:=Sort[Select[Flatten[3^First[#] 7^Last[#] & /@ Tuples[{Range[0, Floor[Log[3, upto]]], Range[0, Floor[Log[7, upto]]]}]], # <= upto &]]; f[120000]  (* Harvey P. Dale, Mar 04 2011 *)
fQ[n_] := PowerMod[21, n, n] == 0; Select[Range[120000], fQ] (* Bruno Berselli, Sep 24 2012 *)

PARI

list(lim)=my(v=List(),N);for(n=0,log(lim)\log(7),N=7^n;while(N<=lim,listput(v,N);N*=3));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011

Python

from sympy import integer_log
def A003594(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: 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//7**i,3)[0]+1 for i in range(integer_log(x,7)[0]+1))
    return bisection(f,n,n) # Chai Wah Wu, Sep 16 2024

Formula

The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(21*n)*x^n/(1 - x^n), where mu(n) is the Möbius function A008683. Cf. with the formula of Hanna in A051037. - Peter Bala, Mar 18 2019

Sum_{n>=1} 1/a(n) = (3*7)/((3-1)*(7-1)) = 7/4. - Amiram Eldar, Sep 22 2020

a(n) ~ exp(sqrt(2*log(3)*log(7)*n)) / sqrt(21). - Vaclav Kotesovec, Sep 22 2020

a(n) = 3^A025642(n) * 7^A025665(n). - R. J. Mathar, Jul 06 2025