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%I #53 Oct 29 2023 01:42:21
%S 1,3,7,9,21,27,49,63,81,147,189,243,343,441,567,729,1029,1323,1701,
%T 2187,2401,3087,3969,5103,6561,7203,9261,11907,15309,16807,19683,
%U 21609,27783,35721,45927,50421,59049,64827,83349,107163,117649
%N Numbers of the form 3^i*7^j with i, j >= 0.
%H Reinhard Zumkeller, <a href="/A003594/b003594.txt">Table of n, a(n) for n = 1..10000</a> (first 70 terms from Vincenzo Librandi)
%H Vaclav Kotesovec, <a href="/A003594/a003594.jpg">Graph - the asymptotic ratio (600000 terms)</a>.
%F 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
%F Sum_{n>=1} 1/a(n) = (3*7)/((3-1)*(7-1)) = 7/4. - _Amiram Eldar_, Sep 22 2020
%F a(n) ~ exp(sqrt(2*log(3)*log(7)*n)) / sqrt(21). - _Vaclav Kotesovec_, Sep 22 2020
%t 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 *)
%t fQ[n_] := PowerMod[21, n, n] == 0; Select[Range[120000], fQ] (* _Bruno Berselli_, Sep 24 2012 *)
%o (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
%o (Magma) [n: n in [1..120000] | PrimeDivisors(n) subset [3,7]]; // _Bruno Berselli_, Sep 24 2012
%o (Haskell)
%o import Data.Set (singleton, deleteFindMin, insert)
%o a003594 n = a003594_list !! (n-1)
%o a003594_list = f $ singleton 1 where
%o f s = y : f (insert (3 * y) $ insert (7 * y) s')
%o where (y, s') = deleteFindMin s
%o -- _Reinhard Zumkeller_, May 16 2015
%o (GAP) Filtered([1..120000],n->PowerMod(21,n,n)=0); # _Muniru A Asiru_, Mar 19 2019
%Y Cf. A003586, A003591, A003592, A003593, A003595.
%K nonn
%O 1,2
%A _N. J. A. Sloane_
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