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 1-6 of 6 results.

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

A353444 Integers m such that the decimal expansion of 1/m contains the digit 8.

Original entry on oeis.org

7, 12, 14, 17, 19, 23, 26, 28, 29, 31, 34, 35, 38, 42, 43, 46, 47, 48, 49, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 73, 76, 77, 78, 79, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 98, 102, 103, 104, 105, 107, 109, 112, 113, 114, 115, 116, 117, 118
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 = 12 is a term since 1/12 = 0.08333333333...
m = 17 is a term since 1/17 = 0.05882352941176470588235294117647...
m = 125 is a term since 1/125 = 0.008.

Links

Crossrefs

A351474 (largest digit=8) and A352161 (smallest digit=8) are subsequences.
Similar with digit k: A352154 (k=0), A353437 (k=1), A353438 (k=2), A353439 (k=3), A353440 (k=4), A353441 (k=5), A353442 (k=6), A353443 (k=7), this sequence (k=8), A333237 (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 150, MemberQ[f@#, 8] &]
Select[Range[150],MemberQ[realDigitsRecip[#],8]&] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jan 11 2025 *)

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

Views

Author

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

Keywords

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).

Examples

			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.

Links

Crossrefs

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).

Programs

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

Views

Author

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

Keywords

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.

Examples

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

Links

Crossrefs

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.

Programs

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

Views

Author

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

Keywords

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.

Examples

			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.

Crossrefs

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.

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 6 &]
  • 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
A351472_list = list(islice(A351472_gen(), 20)) # Chai Wah Wu, Feb 17 2022

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

Views

Author

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

Keywords

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}.

Examples

			As 1/37 = 0.027027027..., 37 is a term.
As 1/148 = 0.00675675675675..., 148 is a term.

Links

Crossrefs

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.

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 7 &]
  • 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<A351473_list = list(islice(A351473_gen(),20)) # Chai Wah Wu, May 02 2023
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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