| Displaying 1-8 of 8 results found.
page
1
Numbers m such that the decimal expansion of 1/m contains the digit 0, ignoring leading and trailing 0's.
+10
17
11, 13, 17, 19, 21, 23, 27, 29, 31, 33, 34, 37, 38, 39, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 57, 58, 59, 61, 62, 63, 67, 68, 69, 71, 73, 76, 77, 78, 79, 81, 83, 84, 85, 86, 87, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart 1 ( A003592) would be terms. If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive.
Some subsequences:
{11, 111, 1111, ...} = A002275 \ {0, 1} {33, 333, 3333, ...} = A002277 \ {0, 3}. {77, 777, 7777, ...} = A002281 \ {0, 7} {11, 101, 1001, 10001, ...} = A000533 \ {1}.
EXAMPLE
m = 13 is a term since 1/13 = 0.0769230769230769230... has a periodic part = '07692307' or '76923070' with a 0.
m = 14 is not a term since 1/14 = 0.0714285714285714285... has a periodic part = '714285' which has no 0 (the only 0 is a leading 0).
MAPLE
removeInitial0:= proc(L) local i;
for i from 1 to nops(L) do if L[i] <> 0 then return L[i..-1] fi od;
[]
end proc:
filter:= proc(n) local q;
q:= NumberTheory:-RepeatingDecimal(1/n);
member(0, removeInitial0(NonRepeatingPart(q))) or member(0, RepeatingPart(q))
end proc:
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 200, Min@ f@# == 0 &]
Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.
+10
8
1, 6, 7, 8, 9, 10, 14, 24, 26, 28, 32, 35, 54, 55, 56, 60, 64, 65, 66, 70, 72, 74, 75, 80, 82, 88, 90, 100, 104, 112, 128, 140, 175, 176, 224, 240, 260, 280, 320, 350, 432, 448, 468, 504, 512, 528, 540, 548, 550, 560, 572, 576, 584, 592, 600, 616, 625, 640, 650, 660
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.
{8, 88, 888, ...} = A002282 \ {0} is a subsequence.
EXAMPLE
m = 14 is a term since 1/14 = 0.0714285714285714285... and the smallest term after the leading 0 is 1.
m = 240 is a term since 1/240 = 0.00416666666... and the smallest term after the leading 0's is 1.
m = 888 is a term since 1/888 = 0.001126126126... and the smallest term after the leading 0's is 1.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 1 &]
PROG
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352155_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, n), multiplicity(5, n)
k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
c = k//n
s = str(m*k//n-c*m).zfill(t)
if s == '0' and min(str(c)) == '1':
yield n
elif '0' not in s and min(str(c).lstrip('0')+s) == '1':
yield n
Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.
+10
8
4, 5, 16, 36, 40, 44, 45, 50, 108, 160, 216, 252, 288, 292, 308, 360, 364, 375, 396, 400, 404, 440, 444, 450, 500, 1024, 1080, 1375, 1600, 2072, 2160, 2368, 2520, 2880, 2920, 3080, 3125, 3375, 3600, 3640, 3750, 3848, 3960, 4000, 4040, 4125, 4224, 4368, 4400, 4440, 4500, 5000
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms.
EXAMPLE
m = 16 is a term since 1/16 = 0.0625 and the smallest term after the leading 0 is 2.
m = 216 is a term since 1/216 = 0.004629629629... and the smallest term after the leading 0's is 2.
m = 4444 is not a term since 1/4444 = 0.00022502250225... and the smallest term after the leading 0's is 0.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 2 &]
PROG
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352156_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, n), multiplicity(5, n)
k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
c = k//n
s = str(m*k//n-c*m).zfill(t)
if s == '0' and min(str(c)) == '2':
yield n
elif '0' not in s and min(str(c).lstrip('0')+s) == '2':
yield n
Numbers m such that the smallest digit in the decimal expansion of 1/m is 4, ignoring leading and trailing 0's.
+10
8
22, 25, 144, 220, 225, 250, 1056, 1184, 1440, 2184, 2200, 2250, 2500, 10560, 11840, 14400, 15625, 20625, 21024, 21840, 22000, 22500, 25000, 104192, 105600, 115625, 118400, 144000, 156250, 168192, 179712, 206250, 210240, 213312, 218400, 220000
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms: 22, 25, 144, 225, 1056, 1184, ...
EXAMPLE
m = 22 is a term since 1/22 = 0.045454545... and the smallest digit after the leading 0 is 4.
m = 1184 is a term since 1/1184 = 0.00084459459... and the smallest digit after the leading 0's is 4.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 4 &]
PROG
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352158_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, n), multiplicity(5, n)
k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
c = k//n
s = str(m*k//n-c*m).zfill(t)
if s == '0' and min(str(c)) == '4':
yield n
elif '0' not in s and min(str(c).lstrip('0')+s) == '4':
yield n
Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.
+10
8
2, 18, 20, 132, 148, 180, 200, 1320, 1480, 1800, 2000, 13008, 13200, 14544, 14800, 18000, 20000, 130080, 132000, 145440, 148000, 180000, 200000, 1300800, 1320000, 1454400, 1480000, 1734375, 1800000, 2000000, 11521152, 12890625, 13008000, 13200000, 14544000, 14800000
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If k is a term, 10*k is also a term; so, terms with no trailing zeros are all primitive terms: 2, 18, 132, 148, 14544, ...
EXAMPLE
m = 148 is a term since 1/148 = 0.00675675675... and the smallest digit after the leading 0's is 5.
m = 1320 is a term since 1/1320 = 0.000075757575... and the smallest digit after the leading 0's is 5.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 5 &]
PROG
(PARI) is(n) = my(d=#digits(n-1), m=9, r=10^d, x=valuation(n, 2), y=valuation(n, 5)); for(k=1, max(x, y)-d*(n==x=2^x*5^y)+znorder(Mod(10, n/x)), if(5>m=min(m, r\n), return(0)); r=r%n*10); m==5; \\ Jinyuan Wang, Mar 27 2022 (Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352159_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, n), multiplicity(5, n)
k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
c = k//n
s = str(m*k//n-c*m).zfill(t)
if s == '0' and min(str(c)) == '5':
yield n
elif '0' not in s and min(str(c).lstrip('0')+s) == '5':
yield n
Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 6, ignoring leading and trailing 0's.
+10
8
15, 150, 1500, 15000, 103125, 150000, 1031250, 1500000, 10312500, 15000000, 103125000, 130078125, 150000000, 1031250000, 1300781250, 1500000000, 10312500000
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term (see examples).
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If t is a term, 10*t is also a term; so, terms with no trailing zeros are all primitive terms: 15, 103125, 130078125, ...
Note that for k = 7, if any term exists, it must be greater than 10^10. - Jinyuan Wang, Mar 28 2022
EXAMPLE
m = 150 is a term since 1/150 = 0.0066666666... and the smallest digit after the leading 0's is 6.
m = 103125 is a term since 1/103125 = 0.000009696969... and the smallest digit after the leading 0's is 6.
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 6 &]
PROG
(Python)
from itertools import count, islice
from sympy import multiplicity, n_order
def A352160_gen(startvalue=1): # generator of terms >= startvalue for n in count(max(startvalue, 1)):
m2, m5 = multiplicity(2, n), multiplicity(5, n)
k, m = 10**max(m2, m5), 10**(t := n_order(10, n//2**m2//5**m5))-1
c = k//n
s = str(m*k//n-c*m).zfill(t)
if '0' not in s and min(str(c).lstrip('0')+s) == '6':
yield n
Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 8, ignoring leading and trailing 0's.
+10
8
125, 1125, 1250, 11250, 12500, 112500, 125000, 1125000, 1250000, 11250000, 12500000, 112500000, 125000000, 1125000000, 1250000000
COMMENTS
Leading 0's are not considered, otherwise every integer >= 11 would be a term.
Trailing 0's are also not considered, otherwise numbers of the form 2^i*5^j with i, j >= 0, apart from 1 ( A003592) would be terms. If t is a term, 10*t is also a term; so, terms with no trailing zeros are all primitive terms: 125, 1125, ...
Note that for k = 7, if any term exists, it must be greater than 10^10. - Jinyuan Wang, Mar 29 2022
EXAMPLE
m = 125 is a term since 1/125 = 0.008 and the smallest digit after the leading 0's is 8.
m = 1125 is a term since 1/1125 = 0.00088888888... and the smallest digit after the leading 0's is 8.
Integers m such that the decimal expansion of 1/m contains the digit 3.
+10
7
3, 12, 13, 17, 19, 23, 26, 27, 28, 29, 30, 31, 32, 33, 34, 38, 41, 42, 43, 46, 47, 48, 49, 51, 52, 53, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 81, 83, 85, 87, 88, 89, 92, 93, 94, 95, 97, 98, 102, 103, 104, 105, 106, 107, 109, 113, 114, 115, 116
COMMENTS
If m is a term, 10*m is also a term, so terms with no trailing zeros are all primitive terms.
EXAMPLE
m = 12 is a term since 1/12 = 0.083333333333... (here, 3 is the smallest digit).
m = 13 is a term since 1/13 = 0.076923076923...
m = 75 is a term since 1/15 = 0.013333333333... (here, 3 is the largest digit).
MATHEMATICA
f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 125, MemberQ[f@#, 3] &]
CROSSREFS
A350814 (largest digit=3) and A352157 (smallest digit=3) are subsequences.
Search completed in 0.031 seconds
|