A333402 -id:A333402 - cp's OEIS frontend

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

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

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

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}.

			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).

Robert Israel, Table of n, a(n) for n = 1..10000

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.

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

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 200, Min@ f@# == 0 &]

A352153(a(n)) = 0.

A333236 Largest digit in the decimal expansion of 1/n.

Original entry on oeis.org

1, 5, 3, 5, 2, 6, 8, 5, 1, 1, 9, 8, 9, 8, 6, 6, 9, 5, 9, 5, 9, 5, 9, 6, 4, 8, 7, 8, 9, 3, 9, 5, 3, 9, 8, 7, 7, 9, 6, 5, 9, 9, 9, 7, 2, 9, 9, 8, 9, 2, 9, 9, 9, 8, 8, 8, 9, 9, 9, 6, 9, 9, 8, 6, 8, 5, 9, 9, 9, 8, 9, 8, 9, 5, 3, 9, 9, 8, 8, 5, 9, 9, 9, 9, 9, 9, 9, 6, 9, 1, 9, 9, 8, 9, 9, 6, 9, 9, 1, 1, 9, 9, 9, 9, 9

Offset: 1

Andrew Slattery, Mar 12 2020

			a(50) = 2 because the largest digit in 1/50 = 0.02 is 2.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333237 (a(n) = 9), A333402 (a(n) = 1).

a[n_] := Max@RealDigits[1/n][[1]]; Array[a, 88] (* Giovanni Resta, Mar 12 2020 *)

from sympy import n_order, multiplicity
def A333236(n):
    m2, m5 = multiplicity(2,n), multiplicity(5,n)
    return int(max(str(10**(max(m2,m5)+n_order(10,n//2**m2//5**m5))//n))) # Chai Wah Wu, Feb 07 2022

a(n) = max_{i=0..n} (floor(10^i/n) mod 10).

a(10*n) = a(n) and a(n) = n iff n = 1, 3, 6. - Bernard Schott, Mar 19 2020

More terms from Giovanni Resta, Mar 12 2020

A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Feb 10 2021

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

			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.

Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Similar with largest digit k: A333402 (k=1), A333237 (k=9).
Subsequence: A093143 \ {1}.
Decimal expansion: A021499 (1/495), A021823 (1/819).

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 2 &] (* Amiram Eldar, Feb 10 2021 *)

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
A341383_list = list(islice(A341383_gen(),10)) # Chai Wah Wu, Feb 07 2022

Missing terms added by Amiram Eldar, Feb 10 2021

A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Jan 30 2022

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, 30, 300, ...} = A093138 \ {1}.
{3, 33, 333, ...} = A002277 \ {0}.
{3, 33, 303, 3003, ...} = 3 * A000533.
{3, 303, 30303, 3030303, ...} = 3 * A094028.

			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.

Index entries for sequences related to decimal expansion of 1/n

Cf. A000533, A094028, A266385, A333236.
Similar with largest digit k: A333402 (k=1), A341383 (k=2), A333237 (k=9).
Subsequences: A002277 \ {0}, A093138 \ {1}.
Decimal expansion: A010701 (1/3), A010674 (1/33).

Select[Range[10^5], Max[RealDigits[1/#][[1]]] == 3 &] (* Amiram Eldar, Jan 30 2022 *)

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)))
A350814_list = list(islice(A350814_gen(),10)) # Chai Wah Wu, Feb 07 2022

More terms from Amiram Eldar, Jan 30 2022

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

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

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.

			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.

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).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 1 &]

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

A352153(a(n)) = 1.

A353437 Integers m such that the decimal expansion of 1/m contains the digit 1.

Original entry on oeis.org

1, 6, 7, 8, 9, 10, 14, 17, 19, 21, 23, 24, 26, 28, 29, 31, 32, 34, 35, 38, 39, 43, 46, 47, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

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

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

			m = 7 is a term since 1/7 = 0.142857142857... (here, 1 is the smallest digit).
m = 17 is a term since 1/17 = 0.05882352941176470588235294117647...
m = 99 is a term since 1/99 = 0.0101010101... (here, 1 is the largest digit).

A333402 (largest digit=1) and A352155 (smallest digit=1) are subsequences.

Similar with digit k: A352154 (k=0), this sequence (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), A353441 (k=5), A353442 (k=6), A353443 (k=7), A353444 (k=8), A333237 (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 125, MemberQ[f@#, 1] &]

A351470 Numbers m such that the largest digit in the decimal expansion of 1/m is 4.

Original entry on oeis.org

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

Offset: 1

Bernard Schott and Robert G. Wilson v, Feb 12 2022

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).

			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.

Index entries for sequences related to decimal expansion of 1/n.

Cf. A333236.

Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), this sequence (k=4), A351471 (k=5), A351472 (k=6), A351473 (k=7), A351474 (k=8), A333237 (k=9).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]];Select[Range@1500000, Max@ f@# == 4 &]

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
A351470_list = list(islice(A351470_gen(), 10)) # Chai Wah Wu, Feb 14 2022

A351471 Numbers m such that the largest digit in the decimal expansion of 1/m is 5.

Original entry on oeis.org

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

Offset: 1

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

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.

			As 1/8 = 0.125, 8 is a term.
As 1/4649 = 0.000215121512151..., 4649 is a term.

Index entries for sequences related to decimal expansion of 1/n.

Subsequences: A002276, A002280.
Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), A351470 (k=4), this sequence (k=5), A351472 (k=6), A351473 (k=7), A351474 (k=8), A333237 (k=9).
Cf. A333236.

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 5 &]

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
A351471_list = list(islice(A351471_gen(), 10)) # Chai Wah Wu, Feb 15 2022

A351472 Numbers m such that the largest digit in the decimal expansion of 1/m is 6.

Original entry on oeis.org

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

Offset: 1

Bernard Schott and Robert G. Wilson v, Feb 16 2022

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.

			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.

Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), A351470 (k=4), A351471 (k=5), this sequence (k=6), A351473 (k=7), A351474 (k=8), A333237 (k=9).

Cf. A094028, A333236.

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 6 &]

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
A351472_list = list(islice(A351472_gen(), 20)) # Chai Wah Wu, Feb 17 2022

A351474 Numbers m such that the largest digit in the decimal expansion of 1/m is 8.

Original entry on oeis.org

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

Offset: 1

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

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).

			As 1/7 = 0.142857142857142857..., 7 is a term.
As 1/26 = 0.0384615384615384615..., 26 is another term.

Index entries for sequences related to decimal expansion of 1/n.

Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), A351470 (k=4), A351471 (k=5), A351472 (k=6), A351473 (k=7), this sequence (k=8), A333237 (k=9).
Cf. A333236.
Decimal expansion of: A020806 (1/7), A021058 (1/54), A021060 (1/56), A021067 (1/63), A021069 (1/65), A021083 (1/79), A021097 (1/93).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 8 &]

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

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<A351474_list = list(islice(A351474_gen(),20)) # Chai Wah Wu, May 02 2023

A333236(a(n)) = 8.

cp's OEIS Frontend

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

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A333236 Largest digit in the decimal expansion of 1/n.

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A341383 Numbers m such that the largest digit in the decimal expansion of 1/m is 2.

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A350814 Numbers m such that the largest digit in the decimal expansion of 1/m is 3.

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

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A353437 Integers m such that the decimal expansion of 1/m contains the digit 1.

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A351470 Numbers m such that the largest digit in the decimal expansion of 1/m is 4.

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