%I #102 Aug 01 2024 01:19:46

%S 1,2,4,5,8,10,16,20,25,32,40,50,64,80,100,125,128,160,200,250,256,320,

%T 400,500,512,625,640,800,1000,1024,1250,1280,1600,2000,2048,2500,2560,

%U 3125,3200,4000,4096,5000,5120,6250,6400,8000,8192,10000,10240,12500,12800

%N Numbers of the form 2^i*5^j with i, j >= 0.

%C These are the natural numbers whose reciprocals are terminating decimals. - _David Wasserman_, Feb 26 2002

%C 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

%C Where record values greater than 1 occur in A165706: A165707(n) = A165706(a(n)). - _Reinhard Zumkeller_, Sep 26 2009

%C 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

%C A204455(5*a(n)) = 5, and only for these numbers. - _Wolfdieter Lang_, Feb 04 2012

%C 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

%C 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

%D Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See p. 73.

%H Reinhard Zumkeller, <a href="/A003592/b003592.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A003592/a003592.jpg">Graph - the asymptotic ratio (200000 terms)</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RegularNumber.html">Regular Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DecimalExpansion.html">Decimal Expansion</a>

%F 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

%F a(n) ~ exp(sqrt(2*log(2)*log(5)*n)) / sqrt(10). - _Vaclav Kotesovec_, Sep 22 2020

%p isA003592 := proc(n)

%p if n = 1 then

%p true;

%p else

%p return (numtheory[factorset](n) minus {2,5} = {} );

%p end if;

%p end proc:

%p A003592 := proc(n)

%p option remember;

%p if n = 1 then

%p 1;

%p else

%p for a from procname(n-1)+1 do

%p if isA003592(a) then

%p return a;

%p end if;

%p end do:

%p end if;

%p end proc: # _R. J. Mathar_, Jul 16 2012

%t twoFiveableQ[n_] := PowerMod[10, n, n] == 0; Select[Range@ 10000, twoFiveableQ] (* _Robert G. Wilson v_, Jan 12 2012 *)

%t 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 *)

%t maxExpo = 14; Sort@ Flatten@ Table[2^i * 5^j, {i, 0, maxExpo}, {j, 0, Log[5, 2^(maxExpo - i)]}] (* Or *)

%t Union@ Flatten@ NestList[{2#, 4#, 5#} &, 1, 7] (* _Robert G. Wilson v_, Apr 16 2011 *)

%o (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

%o (Sage)

%o def isA003592(n) :

%o return not any(d != 2 and d != 5 for d in prime_divisors(n))

%o @CachedFunction

%o def A003592(n) :

%o if n == 1 : return 1

%o k = A003592(n-1) + 1

%o while not isA003592(k) : k += 1

%o return k

%o [A003592(n) for n in (1..48)] # _Peter Luschny_, Jul 20 2012

%o (Magma) [n: n in [1..10000] | PrimeDivisors(n) subset [2,5]]; // _Bruno Berselli_, Sep 24 2012

%o (Haskell)

%o import Data.Set (singleton, deleteFindMin, insert)

%o a003592 n = a003592_list !! (n-1)

%o a003592_list = f $ singleton 1 where

%o f s = y : f (insert (2 * y) $ insert (5 * y) s')

%o where (y, s') = deleteFindMin s

%o -- _Reinhard Zumkeller_, May 16 2015

%o (Python)

%o # A003592.py

%o from heapq import heappush, heappop

%o def A003592():

%o pq = [1]

%o seen = set(pq)

%o while True:

%o value = heappop(pq)

%o yield value

%o seen.remove(value)

%o for x in 2*value, 5*value:

%o if x not in seen:

%o heappush(pq, x)

%o seen.add(x)

%o sequence = A003592()

%o A003592_list = [next(sequence) for _ in range(100)]

%o (GAP) Filtered([1..10000],n->PowerMod(10,n,n)=0); # _Muniru A Asiru_, Mar 19 2019

%Y Complement of A085837. Cf. A094958, A022333 (list of j), A022332 (list of i).

%Y Cf. A003586, A003591, A003593, A003594, A003595, A257997.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E Incomplete Python program removed by _David Radcliffe_, Jun 27 2016