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 A003592 Numbers of the form 2^i*5^j with i, j >= 0. 93
 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 625, 640, 800, 1000, 1024, 1250, 1280, 1600, 2000, 2048, 2500, 2560, 3125, 3200, 4000, 4096, 5000, 5120, 6250, 6400, 8000, 8192, 10000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS These are the natural numbers whose reciprocals are terminating decimals. - David Wasserman, Feb 26 2002 A132726(a(n), k) = 0 for k <= a(n); A051626(a(n)) = 0; A132740(a(n)) = 1; A132741(a(n)) = a(n). - Reinhard Zumkeller, Aug 27 2007 Where record values greater than 1 occur in A165706: A165707(n) = A165706(a(n)). - Reinhard Zumkeller, Sep 26 2009 Also numbers that are divisible by neither 10k - 7, 10k - 3, 10k - 1 nor 10k + 1, for all k > 0. - Robert G. Wilson v, Oct 26 2010 A204455(5*a(n)) = 5, and only for these numbers. - Wolfdieter Lang, Feb 04 2012 Since p = 2 and q = 5 are coprime, sum_{n >= 1} 1/a(n) = sum_{i >= 0} sum_{j >= 0} 1/p^i * 1/q^j = sum_{i >= 0} 1/p^i q/(q - 1) = p*q/((p-1)*(q-1)) = 2*5/(1*4) = 2.5. - Franklin T. Adams-Watters, Jul 07 2014 Conjecture: Each positive integer n not among 1, 4 and 12 can be written as a sum of finitely many numbers of the form 2^a*5^b + 1 (a,b >= 0) with no one dividing another. This has been verified for n <= 3700. - Zhi-Wei Sun, Apr 18 2023 LINKS Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 Vaclav Kotesovec, Graph - the asymptotic ratio (200000 terms) Eric Weisstein's World of Mathematics, Regular Number Eric Weisstein's World of Mathematics, Decimal Expansion FORMULA The characteristic function of this sequence is given by Sum_{n >= 1} x^a(n) = Sum_{n >= 1} mu(10*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 a(n) ~ exp(sqrt(2*log(2)*log(5)*n)) / sqrt(10). - Vaclav Kotesovec, Sep 22 2020 MAPLE isA003592 := proc(n) if n = 1 then true; else return (numtheory[factorset](n) minus {2, 5} = {} ); end if; end proc: A003592 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if isA003592(a) then return a; end if; end do: end if; end proc: # R. J. Mathar, Jul 16 2012 MATHEMATICA twoFiveableQ[n_] := PowerMod[10, n, n] == 0; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Jan 12 2012 *) twoFiveableQ[n_] := Union[ MemberQ[{1, 3, 7, 9}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 10]] == {False}; twoFiveableQ = True; Select[Range@ 10000, twoFiveableQ] (* Robert G. Wilson v, Oct 26 2010 *) maxExpo = 14; Sort@ Flatten@ Table[2^i * 5^j, {i, 0, maxExpo}, {j, 0, Log[5, 2^(maxExpo - i)]}] (* Or *) Union@ Flatten@ NestList[{2#, 4#, 5#} &, 1, 7] (* Robert G. Wilson v, Apr 16 2011 *) PROG (PARI) list(lim)=my(v=List(), N); for(n=0, log(lim+.5)\log(5), N=5^n; while(N<=lim, listput(v, N); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 28 2011 (Sage) def isA003592(n) : return not any(d != 2 and d != 5 for d in prime_divisors(n)) @CachedFunction def A003592(n) : if n == 1 : return 1 k = A003592(n-1) + 1 while not isA003592(k) : k += 1 return k [A003592(n) for n in (1..48)] # Peter Luschny, Jul 20 2012 (Magma) [n: n in [1..10000] | PrimeDivisors(n) subset [2, 5]]; // Bruno Berselli, Sep 24 2012 (Haskell) import Data.Set (singleton, deleteFindMin, insert) a003592 n = a003592_list !! (n-1) a003592_list = f \$ singleton 1 where f s = y : f (insert (2 * y) \$ insert (5 * y) s') where (y, s') = deleteFindMin s -- Reinhard Zumkeller, May 16 2015 (Python) # A003592.py from heapq import heappush, heappop def A003592(): pq =  seen = set(pq) while True: value = heappop(pq) yield value seen.remove(value) for x in 2*value, 5*value: if x not in seen: heappush(pq, x) seen.add(x) sequence = A003592() A003592_list = [next(sequence) for _ in range(100)] (GAP) Filtered([1..10000], n->PowerMod(10, n, n)=0); # Muniru A Asiru, Mar 19 2019 CROSSREFS Complement of A085837. Cf. A094958. Cf. A003586, A003591, A003593, A003594, A003595, A257997. Sequence in context: A181666 A067943 A067937 * A192716 A159765 A018653 Adjacent sequences: A003589 A003590 A003591 * A003593 A003594 A003595 KEYWORD nonn,easy AUTHOR N. J. A. Sloane EXTENSIONS Incomplete Python program removed by David Radcliffe, Jun 27 2016 STATUS approved

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Last modified September 24 13:22 EDT 2023. Contains 365579 sequences. (Running on oeis4.)