A083118 -id:A083118 - cp's OEIS frontend

A083117 Smallest k such that k*n contains a single digit with multiplicity, or 0 if no such number exists.

1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 37, 8547, 15873, 37, 0, 65359477124183, 37, 5847953216374269, 0, 37, 1, 48309178743961352657, 37, 0, 8547, 37, 15873, 38314176245210727969348659, 0, 3584229390681, 0, 1, 65359477124183

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 23 2003

a(n) = 0 if n = 10m, 16m or 25m.

Amarnath Murthy, "On the divisors of the Smarandache Unary sequence," Smarandache Notions Journal, Volume 11, 1-2-3, Spring 2000.

Bo Gyu Jeong, Table of n, a(n) for n = 1..1000
Bo Gyu Jeong, C++ code which computes A083116 and this sequence

Cf. A083116, A083118.

from itertools import count
def A083117(n):
    if not (n%10 and n%16 and n%25): return 0
    for l in count(1):
        k = (10**l-1)//9
        for a in range(1,10):
            b, c = divmod(a*k,n)
            if not c:
                return b # Chai Wah Wu, Jan 23 2024

a(21) corrected by Bo Gyu Jeong, Jun 12 2012

More terms from Bo Gyu Jeong, Jun 13 2012

A083116 Smallest multiple of n using a single digit with multiplicity, or 0 if no such number exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 11, 444, 111111, 222222, 555, 0, 1111111111111111, 666, 111111111111111111, 0, 777, 22, 1111111111111111111111, 888, 0, 222222, 999, 444444, 1111111111111111111111111111, 0, 111111111111111, 0, 33, 2222222222222222, 555555

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Apr 23 2003

1. If p is a prime > 5 then there exists a d such that a(p) = concatenation of '1' d times where p = k*d + 1 for some k. a(p)= (10^d -1)/9 < ={10^(p-1)- 1}/9.

2. a(n) = 0 if n = 10k, 16k or 25k.

Amarnath Murthy, "On the divisors of the Smarandache Unary sequence," Smarandache Notions Journal, Volume 11, 1-2-3, Spring 2000.

Bo Gyu Jeong, Table of n, a(n) for n = 1..1000
Bo Gyu Jeong, C++ code which computes A083117 and this sequence

Cf. A083117, A083118.

from itertools import count
def A083116(n):
    if not (n%10 and n%16 and n%25): return 0
    for l in count(1):
        k = (10**l-1)//9
        for a in range(1,10):
            if not (m:=a*k)%n:
                return m # Chai Wah Wu, Jan 23 2024

a(21) corrected by Bo Gyu Jeong, Jun 12 2012

More terms from Bo Gyu Jeong, Jun 13 2012

A305322 Repdigit numbers that are divisible by 3.

Original entry on oeis.org

0, 3, 6, 9, 33, 66, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 3333, 6666, 9999, 33333, 66666, 99999, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 3333333, 6666666, 9999999, 33333333, 66666666, 99999999, 111111111, 222222222

Offset: 1

Kritsada Moomuang, May 30 2018

The terms > 0 are (10^d-1)*k/9 for k=1..9 if d is divisible by 3, and for k=3,6,9 otherwise. - Robert Israel, Jun 01 2018

Repdigit remainders A010785(k) mod 3 have period 27. - Karl-Heinz Hofmann, Nov 11 2023

			111 / 3 = 37;
222 / 3 = 74;
333 / 3 = 111;
444 / 3 = 148;
555 / 3 = 185.

Robert Israel, Table of n, a(n) for n = 1..4996
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,1001,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000).

Cf. A008585, A010785.

Cf. A002279 (divisor 5), A366596 (divisor 7), A083118 (the impossible divisors).

L:= proc(d) if d mod 3 = 0 then [$1..9] else [3,6,9] fi end proc:
0,seq(seq((10^d-1)/9*k,k=L(d)),d=1..9); # Robert Israel, Jun 01 2018

def A010785(n): return (n - 9*((n-1)//9))*(10**((n+8)//9) - 1)//9
def A305322(n):
    d0, d1 = divmod(n-1,15)
    if d1 < 7: return A010785(d0 * 27 + d1 * 3)
    return A010785(d0 * 27 + d1 + 12) # Karl-Heinz Hofmann, Nov 26 2023

From Alois P. Heinz, May 30 2018: (Start)
{ A008585 } intersect { A010785 }.
G.f.: 3*(300*x^20 + 200*x^19 + 100*x^18 + 330*x^17 + 220*x^16 + 110*x^15 + 333*x^14 + 296*x^13 + 259*x^12 + 222*x^11 + 185*x^10 + 148*x^9 + 111*x^8 + 74*x^7 + 37*x^6 + 33*x^5 + 22*x^4 + 11*x^3 + 3*x^2 + 2*x + 1)*x^2 / ((x-1) *(x^2 + x + 1) *(x^4 + x^3 + x^2 + x + 1) *(10*x^5-1) *(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1) *(100*x^10 + 10*x^5 + 1)).
a(n) = 1001*a(n-15) - 1000*a(n-30). (End)
From Karl-Heinz Hofmann, Nov 11 2023: (Start)
a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15)*3)    iff (n-1 <= 6 (mod 15)).
a(n) = A010785(floor((n-1)/15)*27 + ((n-1) mod 15) + 12) iff (n-1 > 6 (mod 15)).
(End)

Name clarified by Felix Fröhlich, Jun 01 2018

A366596 Repdigit numbers that are divisible by 7.

Original entry on oeis.org

0, 7, 77, 777, 7777, 77777, 111111, 222222, 333333, 444444, 555555, 666666, 777777, 888888, 999999, 7777777, 77777777, 777777777, 7777777777, 77777777777, 111111111111, 222222222222, 333333333333, 444444444444, 555555555555, 666666666666, 777777777777

Offset: 1

Kritsada Moomuang, Oct 14 2023

7 divides a repdigit iff it consists of only digit 7, or has length 6*k (for any digit).

Repdigit remainders A010785(k) mod 7 have period 54. - Karl-Heinz Hofmann, Dec 04 2023

Karl-Heinz Hofmann, Table of n, a(n) for n = 1..2329
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,1000001,0,0,0,0,0,0,0,0,0,0,0,0,0,-1000000).

Intersection of A008589 and A010785.
Cf. A002281 (a subsequence).
Cf. A305322 (divisor 3), A002279 (divisor 5), A083118 (the impossible divisors).

r(n) = 10^((n+8)\9)\9*((n-1)%9+1); \\ A010785
lista(nn) = select(x->!(x%7), vector(nn, k, r(k-1))); \\ Michel Marcus, Oct 26 2023

def A366596(n):
    digitlen, digit = (n+12)//14*6, (n+12)%14-4
    if digit < 1: digitlen += digit - 1; digit = 7
    return 10**digitlen // 9 * digit # Karl-Heinz Hofmann, Dec 04 2023

From Karl-Heinz Hofmann, Dec 04 2023: (Start)
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14) + 41),   for (n-2) mod 14 >  4.
a(n) = (10^(6*floor((n-2)/14) + 6)-1)/9*(((n-2) mod 14)-4), for (n-2) mod 14 >  4.
a(n) = A010785(floor((n-2)/14)*54 + ((n-2) mod 14)*9 + 7),  for (n-2) mod 14 <= 4.
a(n) = (10^(6*floor((n-2)/14) + 1 + ((n-2) mod 14))-1)/9*7, for (n-2) mod 14 <= 4.
(End)

A365933 a(n) is the period of the remainders when repdigits are divided by n.

Original entry on oeis.org

1, 9, 27, 9, 9, 27, 54, 9, 81, 9, 18, 27, 54, 54, 27, 9, 144, 81, 162, 9, 54, 18, 198, 27, 9, 54, 243, 54, 252, 27, 135, 9, 54, 144, 54, 81, 27, 162, 54, 9, 45, 54, 189, 18, 81, 198, 414, 27, 378, 9, 432, 54, 117, 243, 18, 54, 162, 252, 522, 27, 540, 135, 162, 9, 54

Offset: 1

Karl-Heinz Hofmann, Nov 07 2023

For n>1: Periods are divisible by 9 (= a full cycle in the sequence of repdigits). a(n)/9 is the period of the remainders when repunits are divided by n. So the digit part of the repdigits has no effect on periods generally. For most n the beginning of the periodic part is always A010785(1). If n is a term of A083118 the periodic part starts later after some initial remainders that do not repeat.

			For n = 6:                Remainders of A010785(1..54) mod n.
A010785( 1...9) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(10..18) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(19..27) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]
So the period is 3*9 = 27. Thus a(n) = 27. And the pattern seen above starts again:
A010785(28..36) mod n:      [1, 2, 3, 4, 5, 0, 1, 2, 3]
A010785(37..45) mod n:      [5, 4, 3, 2, 1, 0, 5, 4, 3]
A010785(46..54) mod n:      [3, 0, 3, 0, 3, 0, 3, 0, 3]

Karl-Heinz Hofmann, Table with additional information.

Cf. A010785, A002275.
Cf. A305322 (divisor 3), A002279 (divisor 5), A366596 (divisor 7).
Cf. A083118 (the impossible divisors).

def A365933(n):
    if n == 1: return 1
    remainders, exponent = [], 1
    while (rem:=(10**exponent // 9 % n)) not in remainders:
        remainders.append(rem); exponent += 1
    return (exponent - remainders.index(rem) - 1) * 9

def A365933(n):
    if n==1: return 1
    a,b,x,y=1,1,1%n,11%n
    while x!=y:
        if a==b:
            a<<=1
            x,b=y,0
        y = (10*y+1)%n
        b+=1
    return 9*b # Chai Wah Wu, Jan 23 2024

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A083117 Smallest k such that k*n contains a single digit with multiplicity, or 0 if no such number exists.

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A083116 Smallest multiple of n using a single digit with multiplicity, or 0 if no such number exists.

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A305322 Repdigit numbers that are divisible by 3.

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Maple

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A366596 Repdigit numbers that are divisible by 7.

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A365933 a(n) is the period of the remainders when repdigits are divided by n.

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