A003592 - OEIS
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
COMMENTS
These are the natural numbers whose reciprocals are terminating decimals. - David Wasserman, Feb 26 2002
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
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.
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:
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;
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
if n == 1 : return 1
while not isA003592(k) : k += 1
return k
(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
(Python)
from heapq import heappush, heappop
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)
A003592_list = [next(sequence) for _ in range(100)]
(GAP) Filtered([1..10000], n->PowerMod(10, n, n)=0); # Muniru A Asiru, Mar 19 2019
EXTENSIONS
Incomplete Python program removed by David Radcliffe, Jun 27 2016