A366596 -id:A366596 - cp's OEIS frontend

A305322 Repdigit numbers that are divisible by 3.

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

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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A305322 Repdigit numbers that are divisible by 3.

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

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