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A108090 Numbers of the form (11^i)*(13^j).

%I A108090 #29 Apr 22 2025 03:59:39
%S A108090 1,11,13,121,143,169,1331,1573,1859,2197,14641,17303,20449,24167,
%T A108090 28561,161051,190333,224939,265837,314171,371293,1771561,2093663,
%U A108090 2474329,2924207,3455881,4084223,4826809,19487171,23030293,27217619
%N A108090 Numbers of the form (11^i)*(13^j).
%H A108090 Reinhard Zumkeller, <a href="/A108090/b108090.txt">Table of n, a(n) for n = 1..10000</a>
%F A108090 Sum_{n>=1} 1/a(n) = (11*13)/((11-1)*(13-1)) = 143/120. - _Amiram Eldar_, Sep 23 2020
%F A108090 a(n) ~ exp(sqrt(2*log(11)*log(13)*n)) / sqrt(143). - _Vaclav Kotesovec_, Sep 23 2020
%t A108090 mx = 3*10^7; Sort@ Flatten@ Table[ 11^i*13^j, {i, 0, Log[11, mx]}, {j, 0, Log[13, mx/11^i]}] (* _Robert G. Wilson v_, Aug 17 2012 *)
%t A108090 fQ[n_]:=PowerMod[143, n, n] == 0; Select[Range[2 10^7], fQ] (* _Vincenzo Librandi_, Jun 27 2016 *)
%o A108090 (Haskell)
%o A108090 import Data.Set (singleton, deleteFindMin, insert)
%o A108090 a108090 n = a108090_list !! (n-1)
%o A108090 a108090_list = f $ singleton (1,0,0) where
%o A108090    f s = y : f (insert (11 * y, i + 1, j) $ insert (13 * y, i, j + 1) s')
%o A108090          where ((y, i, j), s') = deleteFindMin s
%o A108090 -- _Reinhard Zumkeller_, May 15 2015
%o A108090 (Magma) [n: n in [1..10^7] | PrimeDivisors(n) subset [11, 13]]; // _Vincenzo Librandi_, Jun 27 2016
%o A108090 (PARI) list(lim)=my(v=List(),t); for(j=0,logint(lim\=1,13), t=13^j; while(t<=lim, listput(v,t); t*=11)); Set(v) \\ _Charles R Greathouse IV_, Aug 29 2016
%o A108090 (Python)
%o A108090 from sympy import integer_log
%o A108090 def A108090(n):
%o A108090     def bisection(f,kmin=0,kmax=1):
%o A108090         while f(kmax) > kmax: kmax <<= 1
%o A108090         kmin = kmax >> 1
%o A108090         while kmax-kmin > 1:
%o A108090             kmid = kmax+kmin>>1
%o A108090             if f(kmid) <= kmid:
%o A108090                 kmax = kmid
%o A108090             else:
%o A108090                 kmin = kmid
%o A108090         return kmax
%o A108090     def f(x): return n+x-sum(integer_log(x//13**i,11)[0]+1 for i in range(integer_log(x,13)[0]+1))
%o A108090     return bisection(f,n,n) # _Chai Wah Wu_, Mar 25 2025
%Y A108090 Cf. A003586, A003592, A003593, A003591, A003594, A003595, A003596, A003597, A003598, A003599, A107326, A107364, A107466, A108056.
%K A108090 nonn,easy
%O A108090 1,2
%A A108090 Douglas Winston (douglas.winston(AT)srupc.com), Jun 03 2005