A333236 -id:A333236 - cp's OEIS frontend

A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

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

Offset: 1

Andrew Slattery, Mar 12 2020

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

			5 is not in the sequence because 1/5 = 0.2 does not contain any 9s.

Robert Israel, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A333236.
Subsequences (for terms > 1): A000533, A002275, A135577, A252491.
Cf. A216664 (a subsequence).
Cf. A187614.

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:
select(f, [$1..1000]); # Robert Israel, Mar 18 2020

Select[Range[120], MemberQ[ Flatten@ RealDigits[1/#][[1]], 9] &] (* Giovanni Resta, Mar 12 2020 *)

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
A333237_list = list(islice(A333237_gen(), 10)) # Chai Wah Wu, Feb 07 2022

A333236(a(n)) = 9.

More terms from Giovanni Resta, Mar 12 2020

A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 19 2020

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, ...}.

			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.

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

Cf. A333236, A333237 (similar, with 9).
Subsequences: A002283, A011557, A052268.
Subsequences: 9*A000533, 9*A094028, 9*A135577, 9*A261544, 9*A330135.

Select[Range[10^4], Max @ RealDigits[1/#][[1]] == 1 &] (* Amiram Eldar, Mar 19 2020 *)

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
A333402_list = list(islice(A333402_gen(),30)) # Chai Wah Wu, Feb 17 2022

A333236(a(n))= 1.

More terms from Jinyuan Wang, Mar 19 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

A352153 Smallest digit in the decimal expansion of 1/n, ignoring leading and trailing 0's.

Original entry on oeis.org

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

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 06 2022

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.

			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.

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

Cf. A003592, A333236 (largest digit).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Array[Min@ f@# &, 105]

a(n) = n iff n = 1 or n = 3.
a(10*n) = a(n).
a(10^n) = 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.

A351473 Numbers m such that the largest digit in the decimal expansion of 1/m is 7.

Original entry on oeis.org

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

Offset: 1

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

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}.

			As 1/37 = 0.027027027..., 37 is a term.
As 1/148 = 0.00675675675675..., 148 is a 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), this sequence (k=7), A351474 (k=8), A333237 (k=9).

Cf. A073551, A333236.

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 7 &]

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<A351473_list = list(islice(A351473_gen(),20)) # Chai Wah Wu, May 02 2023

cp's OEIS Frontend

A333237 Numbers k such that 1/k contains at least one '9' in its decimal expansion.

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A333402 Numbers m such that the largest digit in the decimal expansion of 1/m is 1.

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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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A352153 Smallest digit in the decimal expansion of 1/n, ignoring leading and trailing 0's.

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

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