A003592 Numbers of the form 2^i*5^j with i, j >= 0.

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

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[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.

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