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 A003592 Numbers of the form 2^i*5^j with i, j >= 0. 104
 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, 10240, 12500, 12800 (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 REFERENCES Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 73. 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[1] = 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 = [1] 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, A022333 (list of j), A022332 (list of i). 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 10 23:44 EDT 2024. Contains 375795 sequences. (Running on oeis4.)