A007732 -id:A007732 - cp's OEIS frontend

A052983 Least multiple of n consisting of a succession of 1's followed by a succession of 0's.

10, 10, 1110, 100, 10, 1110, 1111110, 1000, 1111111110, 10, 110, 11100, 1111110, 1111110, 1110, 10000, 11111111111111110, 1111111110, 1111111111111111110, 100, 1111110, 110, 11111111111111111111110, 111000, 100, 1111110, 1111111111111111111111111110

Offset: 1

Lekraj Beedassy, Jun 26 2003

All entries are differences of two terms of A000042. Since the pigeonhole principle guarantees that, for any m, two among the first m+1 entries of A000042 are congruent modulo m, their difference (i.e. belonging to this sequence) is therefore divisible by m, so that such numbers exist for all m. This sequence is thus infinite.

For n>1, a(n) consists of s 1's and t 0's, where s=A084681(X) and t is the greater of p or q (s=1 for X=1, t=1 for p=q=0), when we write n=X*Y with (X,Y)=1 and Y=2^p*5^q.

			We have a(6)=1110 because 6 divides 1110=6*185, the smallest such one with a string of 1's followed by that of 0's

Cf. A000042, A002371, A007732.

Cf. A084681.

f[n_] := Select[ Map[ FromDigits, IntegerDigits[ Table[ Sum[2^i, {i, k, j, -1}], {j, k, 1, -1}], 2]]/n, IntegerQ[ # ] & ]; g[n_] := Block[{k = 1}, While[ f[n] == {}, k++ ]; n*Min[ f[n]]]; Table[ g[n], {n, 1, 27}]
nn=30;With[{nos=Sort[Flatten[Table[FromDigits[Join[Table[1,{n}], Table[ 0,{i}]]],{n,nn},{i,5}]]]},Flatten[Table[Select[nos,Divisible[#,n]&,1],{n,nn}]]] (* Harvey P. Dale, Mar 09 2014 *)

a(n) = A276348(n) * n; A227362(a(n)) = 10. - Jaroslav Krizek, Aug 30 2016

Edited, corrected and extended by Robert G. Wilson v, Jun 26 2003

A246004 The duodecimal period of 1/n, or 0 if 1/n terminates.

Original entry on oeis.org

0, 0, 0, 0, 4, 0, 6, 0, 0, 4, 1, 0, 2, 6, 4, 0, 16, 0, 6, 4, 6, 1, 11, 0, 20, 2, 0, 6, 4, 4, 30, 0, 1, 16, 12, 0, 9, 6, 2, 4, 40, 6, 42, 1, 4, 11, 23, 0, 42, 20, 16, 2, 52, 0, 4, 6, 6, 4, 29, 4, 15, 30, 6, 0, 4, 1, 66, 16, 11, 12, 35, 0, 36, 9, 20, 6, 6, 2, 26, 4, 0, 40, 41, 6, 16, 42, 4, 1, 8, 4, 6, 11, 30, 23, 12, 0, 16, 42, 1, 20, 100, 16, 102, 2, 12, 52, 53, 0, 54, 4, 9, 6, 112

Offset: 1

Eric Chen, Nov 13 2014

			1/10 = 0.1249724972497... has period 4 in duodecimal, so a(10) = 4.
1/23 = 0.0631694842106316948421... has period 11 in duodecimal, so a(23) = 11.

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

Cf. A007732, A051626 (decimal versions).

Cf. A246489.

a246004[n_Integer] :=
Table[Length[RealDigits[1/i, 12][[1, -1]]], {i,
   n}]; a246004[120] (* Michael De Vlieger, Nov 13 2014, after Harvey P. Dale at A051626 *)

A121090 Period of unit fractions having periodic decimal expansions.

Original entry on oeis.org

1, 1, 6, 1, 2, 1, 6, 6, 1, 16, 1, 18, 6, 2, 22, 1, 6, 3, 6, 28, 1, 15, 2, 16, 6, 1, 3, 18, 6, 5, 6, 21, 2, 1, 22, 46, 1, 42, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1, 6, 22, 15, 46, 18, 1, 96, 42, 2, 4, 16

Offset: 0

Gil Broussard, Aug 11 2006

See A007732, which is the main entry for this sequence.

			The first unit fraction considered is 1/3 because both 1/1 and 1/2 have finite decimal expansions.
a(1) = 1 because 1/3=.33333... whose repeating portion, 3, is of length 1.
Note: 1/4 and 1/5 are skipped because their decimal expansions are finite.
a(2) = 1 because 1/6=.166666... whose repeating portion, 6, is of length 1.
a(3) = 6 because 1/7 =.142857142857... whose repeating portion, 142857, is of length 6.

Cf. A085837, A007732.

a(n) = A007732(A085837(n)). - Kevin Ryde, May 05 2023

A250211 Square array read by antidiagonals: A(m,n) = multiplicative order of m mod n, or 0 if m and n are not coprime.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 4, 1, 1, 0, 2, 0, 4, 0, 1, 1, 1, 0, 1, 2, 0, 3, 1, 1, 0, 1, 0, 0, 0, 6, 0, 1, 1, 1, 2, 2, 1, 2, 3, 2, 6, 1, 1, 0, 0, 0, 4, 0, 6, 0, 0, 0, 1, 1, 1, 1, 1, 4, 1, 2, 2, 3, 4, 10, 1, 1, 0, 2, 0, 2, 0, 0, 0, 6, 0, 5, 0, 1, 1, 1, 0, 2, 0, 0, 1, 2, 0, 0, 5, 0, 12, 1

Offset: 1

Eric Chen, Dec 29 2014

Read by antidiagonals:
m\n   1    2    3    4    5    6    7    8    9    10    11    12    13
1     1    1    1    1    1    1    1    1    1     1     1     1     1
2     1    0    2    0    4    0    3    0    6     0    10     0    12
3     1    1    0    2    4    0    6    2    0     4     5     0     3
4     1    0    1    0    2    0    3    0    3     0     5     0     6
5     1    1    2    1    0    2    6    2    6     0     5     2     4
6     1    0    0    0    1    0    2    0    0     0    10     0    12
7     1    1    1    2    4    1    0    2    3     4    10     2    12
8     1    0    2    0    4    0    1    0    2     0    10     0     4
9     1    1    0    1    2    0    3    1    0     2     5     0     3
10    1    0    1    0    0    0    6    0    1     0     2     0     6
11    1    1    2    2    1    2    3    2    6     1     0     2    12
12    1    0    0    0    4    0    6    0    0     0     1     0     2
13    1    1    1    1    4    1    2    2    3     4    10     1     0
etc.
A(m,n) = Least k>0 such that m^k=1 (mod n), or 0 if no such k exists.
It is easy to prove that column n has period n.
A(1,n) = 1, A(m,1) =1.
If A(m,n) differs from 0, it is period length of 1/n in base m.
The maximum number in column n is psi(n) (A002322(n)), and all numbers in column n (except 0) divide psi(n), and all factors of psi(n) are in column n.
Except the first row, every row contains all natural numbers.

			A(3,7) = 6 because:
3^0 = 1 (mod 7)
3^1 = 3 (mod 7)
3^2 = 2 (mod 7)
3^3 = 6 (mod 7)
3^4 = 4 (mod 7)
3^5 = 5 (mod 7)
3^6 = 1 (mod 7)
...
And the period is 6, so A(3,7) = 6.

Cf. A002322, A111076, A111725, A001918, A008330, A007733, A002326, A007732, A051626, A066799.

See A139366 for another version.

f:= proc(m,n)
  if igcd(m,n) <> 1 then 0
  elif n=1 then 1
  else numtheory:-order(m,n)
  fi
end proc:
seq(seq(f(t-j,j),j=1..t-1),t=2..65); # Robert Israel, Dec 30 2014

a250211[m_, n_] = If[GCD[m, n] == 1, MultiplicativeOrder[m, n], 0]
Table[a250211[t-j, j], {t, 2, 65}, {j, 1, t-1}]

A351396 Composite numbers d such that the period k of the decimal expansion of 1/d is > 1 and divides d-1.

Original entry on oeis.org

33, 55, 91, 99, 148, 165, 175, 246, 259, 275, 325, 370, 385, 451, 481, 495, 496, 505, 561, 592, 656, 657, 703, 715, 825, 909, 925, 1035, 1045, 1105, 1233, 1375, 1476, 1480, 1626, 1729, 1825, 1912, 2035, 2120, 2275, 2368, 2409, 2465, 2475, 2525, 2556, 2752, 2821

Offset: 1

Barry Smyth, Mar 24 2022

For primes p, the period k of the decimal expansion of 1/p divides p-1. This is usually not the case for reciprocals of composites d; instead, the period k always divides phi(d) where phi is Euler's totient function (A000010). This sequence lists the composites d for which k also divides d-1, which satisfies the condition of a pseudoprime, making such composites a sequence of pseudoprimes with respect to the divisibility of d-1 by k.

			33 is a term since 1/33 = 0.030303..., its repetend is 03 so its period is 2, and 2 divides 33-1.
91 is a term since 1/91 = 0.010989010989..., its repetend is 010898 so its period is 6, and 6 divides 91-1.
925000 is a term since 1/925000 = 0.00000108108... has a repetend of 108 and a period of 3, and 3 divides 925000-1.

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..662 from Barry Smyth)
Barry Smyth, Are pseudoprimes hiding out among the composite reciprocals?

Cf. A007732 (digits period), A000010 (totient).

from itertools import count, islice
from sympy import n_order, multiplicity, isprime
def A351396_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda d: not (isprime(d) or (p := n_order(10, d//2**multiplicity(2, d)//5**multiplicity(5, d))) <= 1 or (d-1) % p), count(max(startvalue,1)))
A351396_list = list(islice(A351396_gen(),50)) # Chai Wah Wu, May 19 2022

A158248 Composite numbers with primitive root 10.

Original entry on oeis.org

49, 289, 343, 361, 529, 841, 2209, 2401, 3481, 3721, 4913, 6859, 9409, 11881, 12167, 12769, 16807, 17161, 22201, 24389, 27889, 32041, 32761, 37249, 49729, 52441, 54289, 66049, 69169, 72361, 83521, 97969

Offset: 1

Robert Hutchins, Mar 15 2009

Previous name was: Numbers m whose reciprocal generates a repeating decimal fraction with period phi(m) and m/2 < phi(m) < m-1.
All terms are proper powers of full reptend primes (A001913).
This sequence does not contain every proper power of every term in A001913, for example, A001913 has 487 as its 26th term, but since 10 is not a primitive root of 487^2, 487^2 is not a term of this sequence. - Robert Hutchins, Oct 14 2021
A shorter description appears to be "Composite numbers with primitive root 10". - Arkadiusz Wesolowski, Jul 04 2012 (The two definitions certainly produce the same terms up through 83521. - N. J. A. Sloane, Jul 05 2012)

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A007732, A001913, A046145, A046146.
Subsequence of A244623.
Subsequence of A167797.
Cf. A108989 (for base 2), A346316 (for base 6).

select(n -> not isprime(n) and numtheory:-primroot(9,n) = 10,[$2..10000]);
# N. J. A. Sloane, Jul 05 2012

Select[Range[10^5], GCD[10, #] == 1 && #/2 < MultiplicativeOrder[10, #] < # - 1 &] (* Ray Chandler, Oct 17 2012 *)

More terms from Robert Hutchins, Mar 21 2009
Entry revised by N. J. A. Sloane, Jul 05 2012
New name (using comment by Arkadiusz Wesolowski) from Joerg Arndt, Nov 22 2021

A216415 a(n) = smallest positive m such that 2n-1 | 10^m-1, or 0 if no such m exists.

Original entry on oeis.org

1, 1, 0, 6, 1, 2, 6, 0, 16, 18, 6, 22, 0, 3, 28, 15, 2, 0, 3, 6, 5, 21, 0, 46, 42, 16, 13, 0, 18, 58, 60, 6, 0, 33, 22, 35, 8, 0, 6, 13, 9, 41, 0, 28, 44, 6, 15, 0, 96, 2, 4, 34, 0, 53, 108, 3, 112, 0, 6, 48, 22, 5, 0, 42, 21, 130, 18, 0, 8, 46, 46, 6, 0, 42, 148, 75, 16, 0, 78, 13, 66, 81, 0, 166, 78, 18, 43, 0, 58, 178, 180, 60, 0, 16, 6, 95, 192, 0, 98, 99

Offset: 1

V. Raman, Sep 07 2012

nonn

This is yet another version of the sequences defined in A002329, A007732, A070682, A084680. - N. J. A. Sloane, Sep 08 2012

a(n) gives the multiplicative order of 10 mod (2n-1), if it is finite, or 0 if not defined.

V. Raman, Table of n, a(n) for n = 1..100.

Cf. A002326, A002329, A007732, A070682, A084680. - N. J. A. Sloane, Sep 08 2012

for(i=0,200,i++;if(i%5==0,print1(0","),print1(znorder(Mod(10,i))","))) \\ V. Raman, Nov 22 2012

for(i=0,200,i++;m=0;for(x=1,i,if(((10^x-1))%i==0,m=x;break));print1(m",")) \\ V. Raman, Nov 22 2012

A266385 a(n) = floor(10^k/n) where k is the smallest integer such that the whole first period or the whole terminating fractional part of the decimal expansion of 1/n is shifted to appear before the decimal point in 10^k/n.

Original entry on oeis.org

1, 5, 3, 25, 2, 16, 142857, 125, 1, 1, 9, 83, 76923, 714285, 6, 625, 588235294117647, 5, 52631578947368421, 5, 47619, 45, 434782608695652173913, 416, 4, 384615, 37, 3571428, 344827586206896551724137931, 3, 32258064516129, 3125, 3, 2941176470588235, 285714, 27

Offset: 1

M. F. Hasler, Dec 28 2015

The period is given in A051626 (with 0 if 1/n terminates) and A007732 (with 1 if 1/n terminates). The periodic part is given in A060284 (with initial 0's omitted) and A036275 (with initial 0's appended).

			a(1) = 1 because 1/1 = 1.0 (k = 0),
a(2) = 5 because 1/2 = 0.5 (k = 1),
a(3) = 3 because 1/3 = 0.{3}*, where {...}* means that these digits repeat forever.
a(4) = 25 because 1/4 = 0.25 (k = 2),
a(5) = 2 because 1/5 = 0.2 (k = 1),
a(6) = 16 because 1/6 = 0.1{6}* (k = 2),
a(7) = 142857 because 1/7 = 0.{142857}* (k = 6),
a(8) = 125 because 1/8 = 0.125 (k = 3),
a(9) = 1 because 1/9 = 0.{1}* (k = 1),
a(10) = 1 because 1/10 = 0.1 (k = 1), ...

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

Cf. A036275, A051626, A060284, A007732.

a(n) = A060284(n) (mod 10^A051626(n)).

Name edited and a(13) onwards from Mohammed Yaseen, Jun 03 2021

A334487 a(n) = p(n, 2)*p(n, 5)/p(n, 10) where p(n, b) is the period of repeating digits of 1/n in base b.

Original entry on oeis.org

1, 1, 4, 1, 4, 4, 3, 2, 36, 4, 25, 4, 8, 3, 8, 4, 8, 36, 9, 4, 6, 25, 11, 4, 20, 8, 108, 3, 14, 8, 1, 8, 50, 8, 12, 36, 432, 9, 8, 8, 80, 6, 28, 25, 72, 11, 23, 8, 21, 20, 8, 8, 208, 108, 50, 3, 18, 14, 29, 8, 30, 1, 6, 16, 8, 50, 44, 8, 22, 12, 5, 36, 81, 432, 40

Offset: 1

Michel Marcus, May 03 2020

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A007733, A007736, A007732, A334488.

f[n_, p_] := MultiplicativeOrder[p, n/(p^IntegerExponent[n, p])]; a[n_] := f[n, 2] * f[n, 5] / MultiplicativeOrder[10, n / 2^IntegerExponent[n, 2] / 5^IntegerExponent[n, 5]]; Array[a, 100] (* Amiram Eldar, May 04 2020 *)

a2(n) = znorder(Mod(2,n/2^valuation(n,2))); \\ A007733
a5(n) = znorder(Mod(5,n/5^valuation(n,5))); \\ A007736
a10(n) = znorder(Mod(10,n/2^valuation(n,2)/5^valuation(n,5))); \\ A007732
a(n) = a2(n)*a5(n)/a10(n);

a(n) = A007733(n)*A007736(n)/A007732(n).

A334488 Numbers m for which p(m, 2)*p(m, 5) = p(m, 10), where p(m, b) is the period of repeating digits of 1/m in base b.

Original entry on oeis.org

1, 2, 4, 31, 62, 124, 601, 1202, 2404, 2593, 4808, 5186, 9616, 10372, 18631, 20744, 37262, 41488, 74524, 82976, 149048, 165952, 298096, 331904, 599479, 1198958, 2397916, 204700049, 409400098, 466344409, 668731841, 818800196, 932688818, 1023500245, 1337463682, 1554449047

Offset: 1

Michel Marcus, May 03 2020

Numbers m such that A334487(m) = 1.

Ray Chandler, Table of n, a(n) for n = 1..56 (first 45 terms from Giovanni Resta)
Robert G. Wilson v and Ray Chandler, Known terms (not exhaustive)

Cf. A007733, A007736, A007732, A334487.

a2(n) = znorder(Mod(2,n/2^valuation(n,2))); \\ A007733
a5(n) = znorder(Mod(5,n/5^valuation(n,5))); \\ A007736
a10(n) = znorder(Mod(10,n/2^valuation(n,2)/5^valuation(n,5))); \\ A007732
isok(m) = a2(m)*a5(m) == a10(m);

a(28) from Jinyuan Wang, May 03 2020

a(29)-a(36) from Giovanni Resta, May 04 2020

cp's OEIS Frontend

A052983 Least multiple of n consisting of a succession of 1's followed by a succession of 0's.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A246004 The duodecimal period of 1/n, or 0 if 1/n terminates.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A121090 Period of unit fractions having periodic decimal expansions.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A250211 Square array read by antidiagonals: A(m,n) = multiplicative order of m mod n, or 0 if m and n are not coprime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A351396 Composite numbers d such that the period k of the decimal expansion of 1/d is > 1 and divides d-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

A158248 Composite numbers with primitive root 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A216415 a(n) = smallest positive m such that 2n-1 | 10^m-1, or 0 if no such m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

A266385 a(n) = floor(10^k/n) where k is the smallest integer such that the whole first period or the whole terminating fractional part of the decimal expansion of 1/n is shifted to appear before the decimal point in 10^k/n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions