cp's OEIS Frontend

A025616 Numbers of form 3^i*10^j, with i, j >= 0.

%I A025616 #30 Jul 06 2025 10:01:03
%S A025616 1,3,9,10,27,30,81,90,100,243,270,300,729,810,900,1000,2187,2430,2700,
%T A025616 3000,6561,7290,8100,9000,10000,19683,21870,24300,27000,30000,59049,
%U A025616 65610,72900,81000,90000,100000,177147,196830,218700,243000,270000
%N A025616 Numbers of form 3^i*10^j, with i, j >= 0.
%H A025616 Reinhard Zumkeller, <a href="/A025616/b025616.txt">Table of n, a(n) for n = 1..10000</a>
%F A025616 Sum_{n>=1} 1/a(n) = (3*10)/((3-1)*(10-1)) = 5/3. - _Amiram Eldar_, Sep 25 2020
%F A025616 a(n) ~ exp(sqrt(2*log(3)*log(10)*n)) / sqrt(30). - _Vaclav Kotesovec_, Sep 25 2020
%F A025616 a(n) = 3^A025644(n) *10^A025685(n). - _R. J. Mathar_, Jul 06 2025
%t A025616 n = 10^6; Flatten[Table[3^i*10^j, {i, 0, Log[3, n]}, {j, 0, Log10[n/3^i]}]] // Sort (* _Amiram Eldar_, Sep 25 2020 *)
%o A025616 (Haskell)
%o A025616 import Data.Set (singleton, deleteFindMin, insert)
%o A025616 a025616 n = a025616_list !! (n-1)
%o A025616 a025616_list = f $ singleton (1,0,0) where
%o A025616    f s = y : f (insert (3 * y, i + 1, j) $ insert (10 * y, i, j + 1) s')
%o A025616          where ((y, i, j), s') = deleteFindMin s
%o A025616 -- _Reinhard Zumkeller_, May 15 2015
%o A025616 (PARI) list(lim)=my(v=List(), N); for(n=0, logint(lim\=1, 10), N=10^n; while(N<=lim, listput(v, N); N*=3)); Set(v) \\ _Charles R Greathouse IV_, Jan 10 2018
%o A025616 (Python)
%o A025616 from sympy import integer_log
%o A025616 def A025616(n):
%o A025616     def bisection(f,kmin=0,kmax=1):
%o A025616         while f(kmax) > kmax: kmax <<= 1
%o A025616         kmin = kmax >> 1
%o A025616         while kmax-kmin > 1:
%o A025616             kmid = kmax+kmin>>1
%o A025616             if f(kmid) <= kmid:
%o A025616                 kmax = kmid
%o A025616             else:
%o A025616                 kmin = kmid
%o A025616         return kmax
%o A025616     def f(x): return n+x-sum(integer_log(x//10**i,3)[0]+1 for i in range(integer_log(x,10)[0]+1))
%o A025616     return bisection(f,n,n) # _Chai Wah Wu_, Mar 25 2025
%Y A025616 Cf. A025612, A025621, A025625, A025629, A025632, A025634, A025635, A108761, A003596, A003597, A107988, A003598, A108698, A003599, A107788, A108687, A108779, A108090.
%K A025616 easy,nonn
%O A025616 1,2
%A A025616 _David W. Wilson_