A003592 -id:A003592 - cp's OEIS frontend

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

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

Bernard Schott and Robert G. Wilson v, Mar 19 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: 3, 12, 264, 275, 296, 1875, ...

			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.

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).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 3 &]

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

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

Bernard Schott and Robert G. Wilson v, Mar 24 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: 22, 25, 144, 225, 1056, 1184, ...

			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.

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).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 4 &]

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

A352153(a(n)) = 4.

A352159 Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.

Original entry on oeis.org

2, 18, 20, 132, 148, 180, 200, 1320, 1480, 1800, 2000, 13008, 13200, 14544, 14800, 18000, 20000, 130080, 132000, 145440, 148000, 180000, 200000, 1300800, 1320000, 1454400, 1480000, 1734375, 1800000, 2000000, 11521152, 12890625, 13008000, 13200000, 14544000, 14800000

Offset: 1

Bernard Schott and Robert G. Wilson v, Mar 25 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: 2, 18, 132, 148, 14544, ...

			m = 148 is a term since 1/148 = 0.00675675675... and the smallest digit after the leading 0's is 5.
m = 1320 is a term since 1/1320 = 0.000075757575... and the smallest digit after the leading 0's is 5.

Cf. A351471.
Subsequence: A093136 \ {0}.
Similar with smallest digit k: A352154 (k=0), A352155 (k=1), A352156 (k=2), A352157 (k=3), A352158 (k=4), this sequence (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@# == 5 &]

is(n) = my(d=#digits(n-1), m=9, r=10^d, x=valuation(n, 2), y=valuation(n, 5)); for(k=1, max(x,y)-d*(n==x=2^x*5^y)+znorder(Mod(10, n/x)), if(5>m=min(m, r\n), return(0)); r=r%n*10); m==5; \\ Jinyuan Wang, Mar 27 2022

from itertools import count,islice
from sympy import multiplicity,n_order
def A352159_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)) == '5':
            yield n
        elif '0' not in s and min(str(c).lstrip('0')+s) == '5':
                yield n
A352159_list = list(islice(A352159_gen(),10)) # Chai Wah Wu, Mar 28 2022

A352153(a(n)) = 5.

More terms from Jinyuan Wang, Mar 27 2022

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

Bernard Schott and Robert G. Wilson v, Mar 27 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 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

			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.

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

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).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[ Range@ 1100, Min@ f@# == 6 &]

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

A352153(a(n)) = 6.

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

Bernard Schott, Mar 29 2022

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

			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.

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

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).

A352153(a(n)) = 8.

a(9)-a(15) from Jinyuan Wang, Mar 29 2022

A077313 Primes of the form 2^r*5^s - 1.

Original entry on oeis.org

3, 7, 19, 31, 79, 127, 199, 499, 1249, 1279, 1999, 4999, 5119, 8191, 12799, 20479, 31249, 49999, 51199, 79999, 81919, 131071, 199999, 524287, 799999, 1249999, 1310719, 3124999, 3276799, 4999999, 7812499, 12499999, 19999999, 20479999

Offset: 1

Amarnath Murthy, Nov 04 2002

nonn

Primes p such that 10^p is divisible by p+1. Primes p whose fractions p/(p+1) are terminating decimals, i.e., primes p such that A158911(p)=0. Primes p such that the prime divisors of p+1 are also prime divisors of the numbers m obtained by the concatenation of p and p+1. For example, for p=19, m = 1920, the prime divisors of 20 are {2, 5} and the prime divisors of 1920 are {2, 3, 5}. - Jaroslav Krizek, Feb 25 2013

For n > 1, all terms are congruent to 1 (mod 6). - Muniru A Asiru, Sep 29 2017

			1250000 = 2*2*2*2*5*5*5*5*5*5*5 and 1250000 - 1 = A000040(96469), therefore 1249999 is a term.
List of (r, s): (2, 0), (3, 0), (2, 1), (5, 0), (4, 1), (7, 0), (3, 2), (2, 3), (1, 4), (8, 1), (4, 3), (3, 4), (10, 1), ...  - _Muniru A Asiru_, Sep 29 2017

Ray Chandler, Table of n, a(n) for n = 1..4222

Cf. A003592, A023509, A077497.

A:=Filtered([1..10^7],IsPrime);;    I:=[5];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A077313:=List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]); # Muniru A Asiru, Sep 29 2017

With[{n = 10^8}, Union@ Select[Flatten@ Table[2^p*5^q - 1, {p, 0, Log[2, n/(1)]}, {q, 0, Log[5, n/(2^p)]}], PrimeQ]] (* Michael De Vlieger, Sep 30 2017 *)

More terms from Reinhard Zumkeller, Nov 15 2002

More terms from Vladeta Jovovic, May 08 2003

A085837 Denominators of unit fractions having non-terminating decimal expansions.

Original entry on oeis.org

3, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 24, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84

Offset: 1

Eric W. Weisstein, Jul 04 2003

Complement of A003592.

			1/3=0.3333..., 1/6=0.16666..., 1/7=0.142857142857..., ...

Eric Weisstein's World of Mathematics, Repeating Decimal

Cf. A003592.

isA085837 := proc(n)
        return (numtheory[factorset](n) minus {2,5} <> {} );
end proc:
A085837 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        for a from procname(n-1)+1 do
            if isA085837(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Jul 16 2012

Select[ Range[84], MatchQ[ RealDigits[1/#], {{_, {}}, 0|-1}] &] (* From Jean-François Alcover, Nov 07 2011 *)
Select[Range[100],Depth[RealDigits[1/#]]>3&] (* Harvey P. Dale, May 28 2015 *)

from sympy import integer_log
def A085837(n):
    def f(x): return n+sum((x//5**i).bit_length() for i in range(integer_log(x,5)[0]+1))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

Definition corrected by Robert Israel, Jul 09 2014

A105115 Numbers k such that the decimal representation of 1/k is neither terminating nor purely repeating.

Original entry on oeis.org

6, 12, 14, 15, 18, 22, 24, 26, 28, 30, 34, 35, 36, 38, 42, 44, 45, 46, 48, 52, 54, 55, 56, 58, 60, 62, 65, 66, 68, 70, 72, 74, 75, 76, 78, 82, 84, 85, 86, 88, 90, 92, 94, 95, 96, 98, 102, 104, 105, 106, 108, 110, 112, 114, 115, 116, 118, 120, 122, 124, 126, 130, 132, 134

Offset: 1

David Wasserman, Apr 07 2005

k is in this sequence iff 1) k is divisible by 2 or 5 and 2) k is also divisible by some prime other than 2 and 5. Contains the numbers that are in neither A003592 nor A045572.

The asymptotic density of this sequence is 3/5. - Amiram Eldar, Mar 26 2021

			22 is a member because 1/22 = .045454545..., which has a 0 before the repeating 45.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A003592, A045572.

f[n_, lim_] := Block[{g, a}, g[x_] := First /@ FactorInteger@ x; a = g@ n; Select[Range@ lim, And[1 < GCD[#, n] < #, Length@ Complement[g@ #, a] >= 1] &]]; f[10, 134] (* Michael De Vlieger, Jun 20 2015 *)

A165706 a(0) = 1, a(n) = a([n/2]) + a([n/5]) for n > 1.

Original entry on oeis.org

1, 2, 3, 3, 4, 5, 5, 5, 6, 6, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22

Offset: 0

Reinhard Zumkeller, Sep 26 2009

nonn

For n>0: A165707(n)=a(A003592(n)) and A165707(m)A003592(n).

R. Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A007731, A083662, A088468, A165704.

a(n)=if(n<3, return(n+1)); a(n\2) + a(n\5) \\ Charles R Greathouse IV, Nov 14 2016

A381801 Irregular triangle read by rows: row n lists the residues r mod n of numbers k such that rad(k) | n, where rad = A007947.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 0, 1, 2, 3, 4, 0, 1, 0, 1, 2, 4, 0, 1, 3, 0, 1, 2, 4, 5, 6, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 0, 1, 0, 1, 2, 4, 7, 8, 0, 1, 3, 5, 6, 9, 10, 12, 0, 1, 2, 4, 8, 0, 1, 0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16, 0, 1, 0, 1, 2, 4, 5, 8, 10, 12, 16

Offset: 1

Michael De Vlieger, Mar 07 2025

Define S(p,n) to be the set of residues r (mod n) taken by the power range of prime divisor p, i.e., {p^m, m >= 1}.
Define T(n) to be the union of the tensor product of distinct terms in S(p,n) for all p|n, where the products are expressed mod n.
Row n of this triangle is T(n), a superset of row n of A381799.
For n > 1, the intersection of row n of this triangle and row n of A038566 is {1}.

			Table of c(n) = A381800(n) and T(n) for select n:
 n  c(n)  T(n)
-----------------------------------------------------------------------------
 1    1   {0}
 2    2   {0, 1}
 3    2   {0, 1}
 4    3   {0, 1, 2}
 5    2   {0, 1}
 6    5   {0, 1, 2, 3, 4}
 8    4   {0, 1, 2, 4}
 9    3   {0, 1, 3}
10    7   {0, 1, 2, 4, 5, 6, 8}
11    2   {0, 1}
12    8   {0, 1, 2, 3, 4, 6, 8, 9}
14    6   {0, 1, 2, 4, 7, 8}
15    8   {0, 1, 3, 5, 6, 9, 10, 12}
16    5   {0, 1, 2, 4, 8}
18   12   {0, 1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 16}
20    9   {0, 1, 2, 4, 5, 8, 10, 12, 16}
24   11   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18}
28    9   {0, 1, 2, 4, 7, 8, 14, 16, 21}
30   19   {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}
36   16   {0, 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, 24, 27, 28, 32}
For n = 10, we have S(2,10) = {1, 2, 4, 6, 8} and S(5,10) = {1, 5}. Therefore we have the following distinct products:
   1  2  4  8  6
   5  0
Hence T(10) = {0, 1, 2, 4, 5, 6, 8}; terms in A003592 belong to these residues (mod 10).
For n = 12, we have S(2,12) = {1, 2, 4, 8} and S(3,12) = {1, 3, 9}. Therefore we have the following distinct products:
   1  2  4  8
   3  6  0
   9
Thus T(12) = {0, 1, 2, 3, 4, 6, 8, 9}, terms in A003586 belong to these residues (mod 12).
For n = 30, we have {1, 2, 4, 8, 16}, {1, 3, 9, 21, 27}, and {1, 5, 25}. Therefore we have the following distinct products:
   1  2  4  8  16         5  10  20         25
   3  6 12 24            15   0
   9 18
  27
Thus T(30) = {0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27}; terms in A051037 belong to these residues (mod 30).

Michael De Vlieger, Table of n, a(n) for n = 1..22906 (rows n = 1..500, flattened)
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..36, showing reduced residues mod n in gray and labeling terms in row n. The number n appears on the left in red italic, and row length A381800(n) in blue at right.
Michael De Vlieger, Plot k in row n at (x,y) = (k,-n), n = 1..5000.

Cf. A007947, A038566, A121998, A162306, A381799, A381800.

Table[Union@ Flatten@ Mod[TensorProduct @@ Map[(p = #; NestWhileList[Mod[p*#, n] &, 1, UnsameQ, All]) &, FactorInteger[n][[All, 1]] ], n], {n, 30}]

Row 1 is {0} since 1 is the empty product and the only number that has zero prime factors is 1, congruent to 0 (mod 1).
Row n begins with {0,1} for n > 1.
For prime p, row p = {0,1}.
For prime power p^m, m > 0, row p = union of {0} and {p^i, i < m}.
Row n is a subset of row n of A121998, considering n in A121998 instead as n mod n = 0.
Row n is a superset of row n of A162306, considering n in A162306 instead as n mod n = 0.

cp's OEIS Frontend

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

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

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A352159 Numbers m such that the smallest digit in the decimal expansion of 1/m is 5, ignoring leading and trailing 0's.

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

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

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A077313 Primes of the form 2^r*5^s - 1.

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A085837 Denominators of unit fractions having non-terminating decimal expansions.

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