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.

Previous Showing 11-19 of 19 results.

A353443 Integers m such that the decimal expansion of 1/m contains the digit 7.

Original entry on oeis.org

7, 13, 14, 17, 19, 21, 23, 27, 28, 29, 34, 35, 36, 37, 38, 43, 44, 46, 47, 49, 51, 52, 53, 56, 57, 58, 59, 61, 63, 67, 68, 69, 70, 71, 76, 77, 79, 81, 83, 84, 85, 86, 87, 89, 92, 93, 94, 95, 97, 98, 102, 103, 107, 109, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 126, 127
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 = 27 is a term since 1/27 = 0.037037037... (here, 7 is the largest digit).

Links

Crossrefs

A351473 (largest digit=7) is a subsequence.
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), this sequence (k=7), A353444 (k=8), A333237 (k=9).

Programs

  • Mathematica
    f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 150, MemberQ[f@#, 7] &]

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

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

Views

Author

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

Keywords

Comments

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).

Examples

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

Programs

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

Formula

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

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

A352023 Primes p such that 1/p does not contain digit '9' in its decimal expansion.

Original entry on oeis.org

2, 3, 5, 7, 37, 79, 239, 4649, 62003, 538987, 35121409, 265371653
Offset: 1

Views

Author

Bernard Schott, Feb 28 2022

Keywords

Comments

Terms a(1)-a(9) and a(10)-a(12) were respectively found by Giovanni Resta and Robert Israel (comments in A333237).
The corresponding largest digit in the decimal expansion of 1/a(n) is A352024(n).
If it exists, a(13) > 2.7*10^8.
a(13) > 1360682471 (with A187614). - Jinyuan Wang, Mar 03 2022
a(13) <= 5363222357, a(14) <= 77843839397. - David A. Corneth, Mar 03 2022

Examples

			The largest digit in the decimal expansion of 1/7 = 0.142857142857... is 8 < 9, hence 7 is a term.

Crossrefs

Subsequence of A187614.
Cf. A004023, A333237, A352024.

Programs

  • Maple
    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:
remove(f, [seq(ithprime(i),i=1..10^5)]); # Robert Israel, Mar 16 2022
  • Mathematica
    Select[Range[10^5], PrimeQ[#] && FreeQ[RealDigits[1/#][[1, 1]], 9] &] (* Amiram Eldar, Feb 28 2022 *)
  • PARI
    isok(p) = if (isprime(p), my(m2=valuation(p, 2), m5=valuation(p, 5)); vecmax(digits(floor(10^(max(m2,m5) + znorder(Mod(10, p/2^m2/5^m5))+1)/p))) < 9); \\ Michel Marcus, Feb 28 2022
  • Python
    from sympy import n_order, nextprime
from itertools import islice
def A352023_gen(): # generator of terms
    yield from (2,3,5)
    p = 7
    while True:
        if '9' not in str(10**(n_order(10, p))//p):
            yield p
        p = nextprime(p)
A352023_list = list(islice(A352023_gen(),9)) # Chai Wah Wu, Mar 03 2022

A352024 Largest digit in the decimal expansion of 1/A352023(n).

Original entry on oeis.org

5, 3, 2, 8, 7, 8, 8, 5, 8, 8, 8, 8
Offset: 1

Views

Author

Bernard Schott, Mar 01 2022

Keywords

Comments

All terms are < 9.
A352023(13) <= 5363222357 and A352023(14) <= 77843839397, in both cases, the corresponding largest digit in the decimal expansion of the inverse is 8.

Examples

			A352023(5) = 37, the largest digit in the decimal expansion of 1/37 = 0.027027027027027... is 7, hence a(5) = 7.

Crossrefs

Cf. A187614, A333236, A333237, A352023.

Formula

a(n) = A333236(A352023(n)). - Amiram Eldar, Mar 02 2022
Previous Showing 11-19 of 19 results.

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