cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A352154 Numbers m such that the decimal expansion of 1/m contains the digit 0, ignoring leading and trailing 0's.

Original entry on oeis.org

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
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 14 2022

Keywords

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}.

Examples

			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).

Links

Crossrefs

Cf. A333402, A341383, A350814.
Similar with smallest digit k: this sequence (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).
Cf. A000533, A002275, A002277, A002281.

Programs

  • 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:
select(filter, [$1..300]); # Robert Israel, Apr 26 2023
  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 200, Min@ f@# == 0 &]

Formula

A352153(a(n)) = 0.

A353441 Integers m such that the decimal expansion of 1/m contains the digit 5.

Original entry on oeis.org

2, 4, 7, 8, 14, 16, 17, 18, 19, 20, 22, 23, 26, 28, 29, 31, 32, 34, 35, 38, 39, 40, 42, 43, 46, 47, 49, 51, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 74, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 102, 103, 104, 105, 106, 107, 108, 109
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Apr 25 2022

Keywords

Comments

If m is a term, 10*m is also a term, so terms with no trailing zeros are all primitive terms.

Examples

			m = 7 is a term since 1/7 = 0.142857142857...
m = 22 is a term since 1/22 = 0.04545454545... (here, 5 is the largest digit).
m = 132 is a term since 1/693 = 0.00757575... (here, 5 is the smallest digit).

Links

Crossrefs

A351471 (largest digit=5) and A352159 (smallest digit=5) are subsequences.
Similar with digit k: A352154 (k=0), A353437 (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), this sequence (k=5), A353442 (k=6), A353443 (k=7), A353444 (k=8), A333237 (k=9).
Complement of A362579.

Programs

  • Maple
    filter:= proc(n) local q;
  q:= NumberTheory:-RepeatingDecimal(1/n);
  member(5,RepeatingPart(q)) or member(5, NonRepeatingPart(q))
end proc:
select(filter, [$1..200]); # Robert Israel, Apr 25 2023
  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 125, MemberQ[f@#, 5] &]
  • Python
    from itertools import count, islice
from sympy import multiplicity, n_order
def A353441_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<A353441_list = list(islice(A353441_gen(),20)) # Chai Wah Wu, May 01 2023

A352155 Numbers m such that the smallest digit in the decimal expansion of 1/m is 1, ignoring leading and trailing 0's.

Original entry on oeis.org

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
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 17 2022

Keywords

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.

Examples

			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.

Crossrefs

Cf. A002282, A333402.
Similar with smallest digit k: A352154 (k=0), this sequence (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 1 &]
  • 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
A352155_list = list(islice(A352155_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 1.

A352156 Numbers m such that the smallest digit in the decimal expansion of 1/m is 2, ignoring leading and trailing 0's.

Original entry on oeis.org

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
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Keywords

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.

Examples

			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.

Crossrefs

Cf. A341383.
Subsequences: A093141 \ {1}, A093143 \ {1}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), this sequence (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 2 &]
  • 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
A352156_list = list(islice(A352156_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 2.

A352157 Numbers m such that the smallest digit in the decimal expansion of 1/m is 3, ignoring leading and trailing 0's.

Original entry on oeis.org

3, 12, 30, 120, 264, 275, 296, 300, 1200, 1875, 2112, 2640, 2664, 2750, 2952, 2960, 3000, 10656, 11808, 12000, 18750, 21120, 22944, 26016, 26400, 26640, 27500, 28125, 29088, 29520, 29600, 30000, 103424, 106560, 106656, 118080, 120000, 156288, 187500, 211200, 229440
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 19 2022

Keywords

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: 3, 12, 264, 275, 296, 1875, ...

Examples

			m = 12 is a term since 1/12 = 0.08333333... and the smallest term after the leading 0 is 3.
m = 264 is a term since 1/264 = 0.003787878... and the smallest term after the leading 0's is 3.

Crossrefs

Cf. A093138 \ {1} (subsequence), A350814.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), this sequence (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 3 &]
  • Python
    from itertools import count, islice
from sympy import multiplicity, n_order
def A352157_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) == '3':
            yield n
A352157_list = list(islice(A352157_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 3.

A352158 Numbers m such that the smallest digit in the decimal expansion of 1/m is 4, ignoring leading and trailing 0's.

Original entry on oeis.org

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
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 24 2022

Keywords

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, ...

Examples

			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.

Crossrefs

Cf. A351470.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), this sequence (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 4 &]
  • 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
A352158_list = list(islice(A352158_gen(),20)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 4.

A352160 Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 6, ignoring leading and trailing 0's.

Original entry on oeis.org

15, 150, 1500, 15000, 103125, 150000, 1031250, 1500000, 10312500, 15000000, 103125000, 130078125, 150000000, 1031250000, 1300781250, 1500000000, 10312500000
Offset: 1

Views

Author

Bernard Schott and Robert G. Wilson v, Mar 27 2022

Keywords

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

Examples

			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.

Links

Crossrefs

Cf. A351472.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), this sequence (k=6), A352153 (no known term for k=7), A352161 (k=8), no term (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 6 &]
  • 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
A352160_list = list(islice(A352160_gen(),5)) # Chai Wah Wu, Mar 28 2022

Formula

A352153(a(n)) = 6.

Extensions

a(9)-a(17) from Jinyuan Wang, Mar 28 2022

A352161 Numbers m such that the smallest digit in the decimal expansion of 1/m is k = 8, ignoring leading and trailing 0's.

Original entry on oeis.org

125, 1125, 1250, 11250, 12500, 112500, 125000, 1125000, 1250000, 11250000, 12500000, 112500000, 125000000, 1125000000, 1250000000
Offset: 1

Views

Author

Bernard Schott, Mar 29 2022

Keywords

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

Examples

			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.

Links

Crossrefs

Cf. A351474.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), A352159 (k=5), A352160 (k=6), A352153 (no known term for k=7), this sequence (k=8), no term (k=9).

Formula

A352153(a(n)) = 8.

Extensions

a(9)-a(15) from Jinyuan Wang, Mar 29 2022
Showing 1-8 of 8 results.

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