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

1,2

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).

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

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 *)

Prog

(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
(Magma) [n: n in [1..120000] | PrimeDivisors(n) subset [3, 7]]; // Bruno Berselli, Sep 24 2012
(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
(GAP) Filtered([1..120000], n->PowerMod(21, n, n)=0); # Muniru A Asiru, Mar 19 2019

Crossrefs

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

Sequence in context: A056745 A057263 A057493 * A014854 A108729 A133027

Adjacent sequences: A003591 A003592 A003593 * A003595 A003596 A003597

Keyword

nonn

Author

N. J. A. Sloane

Status

approved