Displaying 1-10 of 12 results found.
Numbers k such that 1/k contains at least one '9' in its decimal expansion.
+10
20
11, 13, 17, 19, 21, 23, 29, 31, 34, 38, 41, 42, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 67, 68, 69, 71, 73, 76, 77, 81, 82, 83, 84, 85, 86, 87, 89, 91, 92, 94, 95, 97, 98, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118
COMMENTS
Almost every prime appears in this sequence.
Among the first 10000 primes, only 2, 3, 5, 7, 37, 79, 239, 4649, and 62003 do not appear in the sequence. - Giovanni Resta, Mar 13 2020 The next primes not in the sequence are 538987, 35121409, and 265371653. - Robert Israel, Mar 18 2020
EXAMPLE
5 is not in the sequence because 1/5 = 0.2 does not contain any 9s.
MAPLE
f:= proc(n) local m, S, r;
m:= 1; S:= {1};
do
r:= floor(m/n);
if r = 9 then return true fi;
m:= (m - r*n)*10;
if member(m, S) then return false fi;
S:= S union {m};
od
end proc:
MATHEMATICA
Select[Range[120], MemberQ[ Flatten@ RealDigits[1/#][[1]], 9] &] (* Giovanni Resta, Mar 12 2020 *)
PROG
(Python)
from itertools import count, islice
from sympy import n_order, multiplicity
def A333237_gen(startvalue=1): # generator of terms for m in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, m), multiplicity(5, m)
if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '9':
yield m
Numbers m such that the largest digit in the decimal expansion of 1/m is 1.
+10
13
1, 9, 10, 90, 99, 100, 900, 909, 990, 999, 1000, 9000, 9009, 9090, 9900, 9990, 9999, 10000, 90000, 90009, 90090, 90900, 90909, 99000, 99900, 99990, 99999, 100000, 900000, 900009, 900090, 900900, 909000, 909090, 990000, 990099, 999000, 999900, 999990, 999999, 1000000
COMMENTS
If m is a term, 10*m is also a term.
If m is a term then m has only digits {1}, {9}, {1,0} or {9,0} in its decimal representation, but this is not sufficient to be a term (see examples).
Some subsequences below (not exhaustive, see crossrefs):
m = 10^k, k >= 0, hence m is in A011557 = {1, 10, 100, 1000, 10000, ...}; m = 9*10^k, k >= 0, hence m is in A052268 = {9, 90, 900, 9000, 90000, ...}; m = 10^k-1, k >= 1, hence m is in A002283 = {9, 99, 999, 9999, 99999, ...}; m = 9*(10^k+1), k >= 1, hence m is in 9* A000533 = {99, 909, 9009, 90009, ...}; m = 9+100*(100^k-1)/11, k >= 0, hence m is in 9* A094028 = {9, 909, 90909, 9090909, ...}.
EXAMPLE
As 1/101 = 0.009900990099..., 101 is not a term.
As 1/909 = 0.001100110011..., 909 is a term.
As 1/9099 = 0.000109902187..., 9099 is not a term.
As 1/9999 = 0.000100010001..., 9999 is also a term.
MATHEMATICA
Select[Range[10^4], Max @ RealDigits[1/#][[1]] == 1 &] (* Amiram Eldar, Mar 19 2020 *)
PROG
(Python)
from itertools import count, islice
def A333402_gen(startvalue=1): # generator of terms >= startvalue for m in count(max(startvalue, 1)):
k = 1
while k <= m:
k *= 10
rset = {0}
while True:
k, r = divmod(k, m)
if max(str(k)) > '1':
break
else:
if r in rset:
yield m
break
rset.add(r)
k = r
while k <= m:
k *= 10
Numbers m such that the largest digit in the decimal expansion of 1/m is 2.
+10
9
5, 45, 50, 450, 495, 500, 819, 825, 4500, 4545, 4950, 4995, 5000, 8190, 8250, 8325, 45000, 45045, 45450, 47619, 49500, 49950, 49995, 50000, 81819, 81900, 82500, 83250, 83325, 89109, 450000, 450045, 450450, 454500, 454545, 476190, 495000, 499500, 499950, 499995, 500000
COMMENTS
If m is a term, 10*m is also a term.
5 is the only prime up to 2.6*10^8 (comments in A333237). Some subsequences: {45, 4545, 454545, ...}, {45045, 45045045, 45045045045, ...}, {45, 495, 4995, 49995, ...}, {819, 81819, 8181819, ...}, {825, 8325, 83325, 833325...}, ...
The subsequence of terms where 1/m has only digits {0,2} is m = 5* A333402 = 5, 45, 50, etc. A333402 is those t where 1/t has only digits {0,1}, so that 1/(5*t) = 2*(1/t)*(1/10) has digits {0,2}, starting from 1/5 = 0.2. These m are also A333402/2 of the even terms from A333402, since A333402 (like here) is self-similar in that the multiples of 10, divided by 10, are the sequence itself. - Kevin Ryde, Feb 13 2021
EXAMPLE
As 1/45 = 0.0202020202..., 45 is a term.
As 1/825 = 0.0012121212121212...., 825 is a term.
As 1/47619 = 0.000021000021000021..., 47619 is a term.
As 1/4545045 = 0.000000220019824..., 4545045 is not a term.
MATHEMATICA
Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 2 &] (* Amiram Eldar, Feb 10 2021 *)
PROG
(Python)
from itertools import count, islice
from sympy import n_order, multiplicity
def A341383_gen(startvalue=1): # generator of terms >= startvalue for m in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, m), multiplicity(5, m)
if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '2':
yield m
Numbers m such that the largest digit in the decimal expansion of 1/m is 3.
+10
8
3, 30, 33, 75, 300, 303, 330, 333, 429, 750, 813, 3000, 3003, 3030, 3125, 3300, 3330, 3333, 4290, 4329, 7500, 7575, 8130, 30000, 30003, 30030, 30300, 30303, 31250, 33000, 33300, 33330, 33333, 42900, 43290, 46875, 75000, 75075, 75750, 76923, 81103, 81300, 300000
COMMENTS
If m is a term, 10*m is also a term.
3 is the only prime up to 2.6*10^8 (see comments in A333237). Some subsequences:
{3, 33, 303, 3003, ...} = 3 * A000533. {3, 303, 30303, 3030303, ...} = 3 * A094028.
EXAMPLE
As 1/33 = 0.0303030303..., 33 is a term.
As 1/75 = 0.0133333333..., 75 is a term.
As 1/429 = 0.002331002331002331..., 429 is a term.
MATHEMATICA
Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* Amiram Eldar, Jan 30 2022 *)
PROG
(Python)
from fractions import Fraction
from itertools import count, islice
from sympy import n_order, multiplicity
def repeating_decimals_expr(f, digits_only=False):
""" returns repeating decimals of Fraction f as the string aaa.bbb[ccc].
returns only digits if digits_only=True.
"""
a, b = f.as_integer_ratio()
m2, m5 = multiplicity(2, b), multiplicity(5, b)
r = max(m2, m5)
k, m = 10**r, 10**n_order(10, b//2**m2//5**m5)-1
c = k*a//b
s = str(c).zfill(r)
if digits_only:
return s+str(m*k*a//b-c*m)
else:
w = len(s)-r
return s[:w]+'.'+s[w:]+'['+str(m*k*a//b-c*m)+']'
def A350814_gen(startvalue=1): # generator of terms >= startvalue return filter(lambda m:max(repeating_decimals_expr(Fraction(1, m), digits_only=True)) == '3', count(max(startvalue, 1)))
Smallest digit in the decimal expansion of 1/n, ignoring leading and trailing 0's.
+10
8
1, 5, 3, 2, 2, 1, 1, 1, 1, 1, 0, 3, 0, 1, 6, 2, 0, 5, 0, 5, 0, 4, 0, 1, 4, 1, 0, 1, 0, 3, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0
COMMENTS
Leading 0's are not considered, otherwise a(n) would be 0 when n >= 11 (see examples for 13 and 14).
Trailing 0's are not also considered, otherwise when 1/n is a terminating decimal ( A003592), a(n) would be also 0.
FORMULA
a(n) = n iff n = 1 or n = 3.
a(10*n) = a(n).
a(10^n) = 1.
EXAMPLE
1/13 = 0.076923076923076923... with periodic part = '769230' (or '076923'), hence a(13) = 0.
1/14 = 0.0714285714285714285... with periodic part = '714285', hence a(14) = 1.
1/40 = 0.025 hence a(40) = 2.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Array[Min@ f@# &, 105]
Numbers m such that the largest digit in the decimal expansion of 1/m is 4.
+10
6
25, 225, 250, 693, 2250, 2439, 2475, 2500, 3285, 4095, 4125, 6930, 6993, 22500, 22725, 23125, 23245, 24390, 24750, 24975, 25000, 30825, 32850, 40950, 41250, 41625, 42735, 69300, 69375, 69735, 69930, 71225, 225000, 225225, 227250, 231250, 232450, 238095, 243309, 243900, 247500, 249750
COMMENTS
If k is a term, 10*k is also a term.
First few primitive terms are 25, 225, 693, 2439, 2475, 3285, 4095, 4125, ...
There is no prime up to 2.6*10^8 (see comments in A333237).
EXAMPLE
As 1/25 = 0.04, and 25 is the smallest number m such that the largest digit in the decimal expansion of 1/m is 4, so a(1) = 25.
As 1/693 = 0.001443001443001443..., so 693 is a term.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 4 &]
PROG
(Python)
from itertools import count, islice
from sympy import n_order, multiplicity
def A351470_gen(startvalue=1): # generator of terms >= startvalue for m in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, m), multiplicity(5, m)
if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '4':
yield m
Numbers m such that the largest digit in the decimal expansion of 1/m is 5.
+10
6
2, 4, 8, 18, 20, 22, 32, 40, 66, 74, 80, 180, 185, 198, 200, 220, 222, 320, 396, 400, 444, 492, 660, 666, 702, 704, 738, 740, 800, 803, 876, 1800, 1818, 1845, 1848, 1850, 1875, 1912, 1980, 1998, 2000, 2200, 2220, 2222, 2409, 2424, 2466, 2849, 3075, 3200, 3212, 3276, 3960, 3996, 4000
COMMENTS
If k is a term, 10*k is also a term.
First few primitive terms are 2, 4, 8, 18, 22, 32, 66, 74, 185, 198, 222, 396, ...
2 and 4649 are the only primes up to 2.6*10^8 (see comments in A333237). Some subsequences:
{2, 22, 222, 2222, ...} = A002276 \ {0}. {66, 666, 6666, ...} = A002280 \ {0, 6}. {18, 1818, 181818, ...} = 18 * A094028.
EXAMPLE
As 1/8 = 0.125, 8 is a term.
As 1/4649 = 0.000215121512151..., 4649 is a term.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 5 &]
PROG
(Python)
from itertools import count, islice
from sympy import n_order, multiplicity
def A351471_gen(startvalue=1): # generator of terms >= startvalue for m in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, m), multiplicity(5, m)
if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '5':
yield m
Numbers m such that the largest digit in the decimal expansion of 1/m is 6.
+10
6
6, 15, 16, 24, 39, 60, 64, 88, 96, 150, 156, 160, 165, 219, 240, 246, 273, 275, 375, 378, 384, 390, 399, 462, 600, 606, 615, 624, 625, 640, 792, 822, 858, 880, 888, 956, 960, 975, 984, 1500, 1515, 1536, 1554, 1560, 1584, 1596, 1600, 1606, 1626, 1628, 1638, 1650, 1665, 1776, 2145
COMMENTS
If k is a term, 10*k is also a term.
First few primitive terms are 6, 15, 16, 24, 39, 64, 88, 96, 156, 165, ...
There is no prime up to 2.6*10^8 (see comments in A333237). Subsequence: {6, 606, 60606, ...} = 6 * A094028.
EXAMPLE
1/6 = 0.166666..., and 6 is the smallest number m such that the largest digit in the decimal expansion of 1/m is 6, so a(1) = 6.
As 1/39 = 0.025641025641..., 39 is a term.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 6 &]
PROG
(Python)
from itertools import count, islice
from sympy import n_order, multiplicity
def A351472_gen(startvalue=1): # generator of terms >= startvalue for m in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, m), multiplicity(5, m)
if max(str(10**(max(m2, m5)+n_order(10, m//2**m2//5**m5))//m)) == '6':
yield m
Numbers m such that the largest digit in the decimal expansion of 1/m is 8.
+10
6
7, 12, 14, 26, 28, 35, 48, 54, 55, 56, 63, 65, 70, 72, 78, 79, 93, 117, 120, 123, 125, 128, 140, 175, 176, 186, 192, 195, 205, 224, 239, 259, 260, 264, 280, 296, 312, 318, 328, 350, 372, 416, 432, 438, 448, 465, 480, 540, 542, 546, 548, 550, 555, 560, 572, 584, 594, 630, 632, 650, 675
COMMENTS
If k is a term, 10*k is also a term. First few primitive terms are 7, 12, 14, 26, 28, 35, 48, 54, 55, 56, 63, 65, 72, ...
The seven primes up to 2.7*10^8 are 7, 79, 239, 62003, 538987, 35121409, 265371653 (see comments in A333237, example section and Crossrefs).
EXAMPLE
As 1/7 = 0.142857142857142857..., 7 is a term.
As 1/26 = 0.0384615384615384615..., 26 is another term.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 8 &]
PROG
(PARI) isok(m) = my(m2=valuation(m, 2), m5=valuation(m, 5)); vecmax(digits(floor(10^(max(m2, m5) + znorder(Mod(10, m/2^m2/5^m5))+1)/m))) == 8; \\ Michel Marcus, Feb 26 2022 (Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A351474_gen(startvalue=1): # generator of terms >= startvalue for a in count(max(startvalue, 1)):
m2, m5 = (~a&a-1).bit_length(), multiplicity(5, a)
k, m = 10**max(m2, m5), 10**n_order(10, a//(1<<m2)//5**m5)-1
if max(max(str(c:=k//a)), max(str(m*k//a-c*m)))=='8':
yield a
Numbers m such that the largest digit in the decimal expansion of 1/m is 7.
+10
5
27, 36, 37, 44, 132, 135, 148, 234, 270, 288, 292, 297, 308, 315, 360, 364, 369, 370, 404, 407, 440, 468, 576, 616, 636, 657, 707, 728, 756, 808, 864, 1287, 1295, 1313, 1314, 1320, 1332, 1350, 1365, 1375, 1386, 1404, 1408, 1476, 1480, 1485, 1507, 1512, 1752, 1804, 1896
COMMENTS
If k is a term, 10*k is also a term.
First few primitive terms are 27, 36, 37, 44, 132, 135, 148, 234, 288, ...
The unique prime up to 2.6*10^8 is 37 (see comments in A333237 and example). Subsequence: {132, 1332, 13332, ...} = A073551 \ {2, 12}.
EXAMPLE
As 1/37 = 0.027027027..., 37 is a term.
As 1/148 = 0.00675675675675..., 148 is a term.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 7 &]
PROG
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A351473_gen(startvalue=1): # generator of terms >= startvalue for a in count(max(startvalue, 1)):
m2, m5 = (~a&a-1).bit_length(), multiplicity(5, a)
k, m = 10**max(m2, m5), 10**n_order(10, a//(1<<m2)//5**m5)-1
if max(max(str(c:=k//a)), max(str(m*k//a-c*m)))=='7':
yield a
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